Methods and apparatus to estimate audience measurement metrics based on users represented in bloom filter arrays

ABSTRACT

Methods, apparatus, systems, and articles of manufacture to estimate audience measurement metrics based on users represented in Bloom filter arrays are disclosed. An apparatus includes a communications interface to receive a first Bloom filter array from a first computer of a first database proprietor. The first Bloom filter array is representative of first users who accessed media. The first users are registered with the first database proprietor. The first Bloom filter array includes a first array of first elements. Values of respective ones of the first elements are either a 0 or a 1 based on whether quantities of the first users allocated to the respective ones of the first elements are even or odd. The apparatus further includes a Bloom filter array analyzer to estimate a first cardinality for the first Bloom filter array. The first cardinality is indicative of a total number of the first users who accessed the media.

FIELD OF THE DISCLOSURE

This disclosure relates generally to monitoring audiences, and, moreparticularly, to methods and apparatus to estimate audience measurementmetrics based on users represented in Bloom filter arrays.

BACKGROUND

Traditionally, audience measurement entities determine audience exposureto media based on registered panel members. That is, an audiencemeasurement entity (AME) enrolls people who consent to being monitoredinto a panel. The AME then monitors those panel members to determinemedia (e.g., television programs or radio programs, movies, DVDs,advertisements, webpages, streaming media, etc.) exposed to those panelmembers. In this manner, the AME can determine exposure metrics (e.g.,audience size) for different media based on the collected mediameasurement data.

As people are accessing more and more media through digital means (e.g.,via the Internet), it is possible for online publishers and/or databaseproprietors providing such media to track all instances of exposure tomedia (e.g., on a census wide level) rather than being limited toexposure metrics based on audience members enrolled panel members of anAME. However, database proprietors are typically only able to trackmedia exposure pertaining to online activity associated with theplatforms operated by the database proprietors. Where media is deliveredvia multiple different platforms of multiple different databaseproprietors, no single database proprietor will be able to provideexposure metrics across the entire population to which the media wasmade accessible. Furthermore, such database proprietors have an interestin preserving the privacy of their users such that there are limitationson the nature of the exposure metrics such database proprietors arewilling to share with one another and/or an interested third party suchas an AME.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is an example environment to implement a technique for loggingimpressions of accesses to server-based media.

FIGS. 2-5 illustrate the creation of a traditional Bloom filter array.

FIGS. 6-9 illustrate the creation of an example modulo 2 Bloom filterarray in accordance with teachings disclosed herein.

FIG. 10 is a block diagram of an example database proprietor apparatusof any one of the database proprietors of FIG. 1 .

FIG. 11 is a block diagram of an audience measurement entity apparatusof the audience measurement entity of FIG. 1 .

FIG. 12 is a flowchart representative of example machine readableinstructions that may be executed to implement the example databaseproprietor apparatus of FIG. 10 .

FIG. 13 is a flowchart representative of example machine readableinstructions that may be executed to implement the example databaseproprietor apparatus of FIG. 10 .

FIG. 14 is a flowchart representative of example machine readableinstructions that may be executed to implement the example audiencemeasurement entity apparatus of FIG. 11 .

FIG. 15 is a flowchart representative of example machine readableinstructions that may be executed to implement the example audiencemeasurement entity apparatus of FIG. 11 .

FIG. 16 is a block diagram of an example processing platform structuredto execute the example instructions of FIGS. 12 and/or 13 to implementthe example database proprietor apparatus of FIG. 10 .

FIG. 17 is a block diagram of an example processing platform structuredto execute the example instructions of FIGS. 14 and/or 15 to implementthe example audience measurement entity apparatus of FIG. 11 .

In general, the same reference numbers will be used throughout thedrawing(s) and accompanying written description to refer to the same orlike parts. As used herein, connection references (e.g., attached,coupled, connected, and joined) may include intermediate members betweenthe elements referenced by the connection reference and/or relativemovement between those elements unless otherwise indicated. As such,connection references do not necessarily infer that two elements aredirectly connected and/or in fixed relation to each other. As usedherein, stating that any part is in “contact” with another part isdefined to mean that there is no intermediate part between the twoparts.

Unless specifically stated otherwise, descriptors such as “first,”“second,” “third,” etc. are used herein without imputing or otherwiseindicating any meaning of priority, physical order, arrangement in alist, and/or ordering in any way, but are merely used as labels and/orarbitrary names to distinguish elements for ease of understanding thedisclosed examples. In some examples, the descriptor “first” may be usedto refer to an element in the detailed description, while the sameelement may be referred to in a claim with a different descriptor suchas “second” or “third.” In such instances, it should be understood thatsuch descriptors are used merely for identifying those elementsdistinctly that might, for example, otherwise share a same name. As usedherein “substantially real time” refers to occurrence in a nearinstantaneous manner recognizing there may be real world delays forcomputing time, transmission, etc. Thus, unless otherwise specified,“substantially real time” refers to real time+/−1 second.

DETAILED DESCRIPTION

Techniques for monitoring user access to an Internet-accessible media,such as digital television (DTV) media and digital content ratings (DCR)media, have evolved significantly over the years. Internet-accessiblemedia is also known as digital media. In the past, such monitoring wasdone primarily through server logs. In particular, entities servingmedia on the Internet would log the number of requests received fortheir media at their servers. Basing Internet usage research on serverlogs is problematic for several reasons. For example, server logs can betampered with either directly or via zombie programs, which repeatedlyrequest media from the server to increase the server log counts. Also,media is sometimes retrieved once, cached locally and then repeatedlyaccessed from the local cache without involving the server. Server logscannot track such repeat views of cached media. Thus, server logs aresusceptible to both over-counting and under-counting errors.

The inventions disclosed in Blumenau, U.S. Pat. No. 6,108,637, which ishereby incorporated herein by reference in its entirety, fundamentallychanged the way Internet monitoring is performed and overcame thelimitations of the server-side log monitoring techniques describedabove. For example, Blumenau disclosed a technique wherein Internetmedia to be tracked is tagged with monitoring instructions. Inparticular, monitoring instructions are associated with the hypertextmarkup language (HTML) of the media to be tracked. When a clientrequests the media, both the media and the monitoring instructions aredownloaded to the client. The monitoring instructions are, thus,executed whenever the media is accessed, be it from a server or from acache. Upon execution, the monitoring instructions cause the client tosend or transmit monitoring information from the client to a contentprovider site. The monitoring information is indicative of the manner inwhich content was displayed.

In some implementations, an impression request or ping request can beused to send or transmit monitoring information by a client device usinga network communication in the form of a hypertext transfer protocol(HTTP) request. In this manner, the impression request or ping requestreports the occurrence of a media impression at the client device. Forexample, the impression request or ping request includes information toreport access to a particular item of media (e.g., an advertisement, awebpage, an image, video, audio, etc.). In some examples, the impressionrequest or ping request can also include a cookie previously set in thebrowser of the client device that may be used to identify a user thataccessed the media. That is, impression requests or ping requests causemonitoring data reflecting information about an access to the media tobe sent from the client device that downloaded the media to a monitoringentity and can provide a cookie to identify the client device and/or auser of the client device. In some examples, the monitoring entity is anaudience measurement entity (AME) that did not provide the media to theclient and who is a trusted (e.g., neutral) third party for providingaccurate usage statistics (e.g., The Nielsen Company, LLC). Since theAME is a third party relative to the entity serving the media to theclient device, the cookie sent to the AME in the impression request toreport the occurrence of the media impression at the client device is athird-party cookie. Third-party cookie tracking is used by measuremententities to track access to media accessed by client devices fromfirst-party media servers.

There are many database proprietors operating on the Internet. Thesedatabase proprietors provide services to large numbers of subscribers.In exchange for the provision of services, the subscribers register withthe database proprietors. Examples of such database proprietors includesocial network sites (e.g., Facebook, Twitter, My Space, etc.),multi-service sites (e.g., Yahoo!, Google, Axiom, Catalina, etc.),online retailer sites (e.g., Amazon.com, Buy.com, etc.), creditreporting sites (e.g., Experian), streaming media sites (e.g., YouTube,Hulu, etc.), etc. These database proprietors set cookies and/or otherdevice/user identifiers on the client devices of their subscribers toenable the database proprietors to recognize their subscribers when theyvisit their web sites.

The protocols of the Internet make cookies inaccessible outside of thedomain (e.g., Internet domain, domain name, etc.) on which they wereset. Thus, a cookie set in, for example, the facebook.com domain (e.g.,a first party) is accessible to servers in the facebook.com domain, butnot to servers outside that domain. Therefore, although an AME (e.g., athird party) might find it advantageous to access the cookies set by thedatabase proprietors, they are unable to do so.

The inventions disclosed in Mazumdar et al., U.S. Pat. No. 8,370,489,which is incorporated by reference herein in its entirety, enable an AMEto leverage the existing databases of database proprietors to collectmore extensive Internet usage by extending the impression requestprocess to encompass partnered database proprietors and by using suchpartners as interim data collectors. The inventions disclosed inMazumdar accomplish this task by structuring the AME to respond toimpression requests from clients (who may not be a member of an audiencemeasurement panel and, thus, may be unknown to the AME) by redirectingthe clients from the AME to a database proprietor, such as a socialnetwork site partnered with the AME, using an impression response. Sucha redirection initiates a communication session between the clientaccessing the tagged media and the database proprietor. For example, theimpression response received at the client device from the AME may causethe client device to send a second impression request to the databaseproprietor. In response to the database proprietor receiving thisimpression request from the client device, the database proprietor(e.g., Facebook) can access any cookie it has set on the client tothereby identify the client based on the internal records of thedatabase proprietor. In the event the client device corresponds to asubscriber of the database proprietor, the database proprietorlogs/records a database proprietor demographic impression in associationwith the user/client device.

As used herein, an impression is defined to be an event in which a homeor individual accesses and/or is exposed to media (e.g., anadvertisement, content, a group of advertisements and/or a collection ofcontent). In Internet media delivery, a quantity of impressions orimpression count is the total number of times media (e.g., content, anadvertisement, or advertisement campaign) has been accessed by a webpopulation (e.g., the number of times the media is accessed). In someexamples, an impression or media impression is logged by an impressioncollection entity (e.g., an AME or a database proprietor) in response toan impression request from a user/client device that requested themedia. For example, an impression request is a message or communication(e.g., an HTTP request) sent by a client device to an impressioncollection server to report the occurrence of a media impression at theclient device. In some examples, a media impression is not associatedwith demographics. In non-Internet media delivery, such as television(TV) media, a television or a device attached to the television (e.g., aset-top-box or other media monitoring device) may monitor media beingoutput by the television. The monitoring generates a log of impressionsassociated with the media displayed on the television. The televisionand/or connected device may transmit impression logs to the impressioncollection entity to log the media impressions.

A user of a computing device (e.g., a mobile device, a tablet, a laptop,etc.) and/or a television may be exposed to the same media via multipledevices (e.g., two or more of a mobile device, a tablet, a laptop, etc.)and/or via multiple media types (e.g., digital media available online,digital TV (DTV) media temporality available online after broadcast, TVmedia, etc.). For example, a user may start watching the Walking Deadtelevision program on a television as part of TV media, pause theprogram, and continue to watch the program on a tablet as part of DTVmedia. In such an example, the exposure to the program may be logged byan AME twice, once for an impression log associated with the televisionexposure, and once for the impression request generated by a tag (e.g.,census measurement science (CMS) tag) executed on the tablet. Multiplelogged impressions associated with the same program and/or same user aredefined as duplicate impressions. Duplicate impressions are problematicin determining total reach estimates because one exposure via two ormore cross-platform devices may be counted as two or more uniqueaudience members. As used herein, reach is a measure indicative of thedemographic coverage achieved by media (e.g., demographic group(s)and/or demographic population(s) exposed to the media). For example,media reaching a broader demographic base will have a larger reach thanmedia that reached a more limited demographic base. The reach metric maybe measured by tracking impressions for known users (e.g., panelists ornon-panelists) for which an audience measurement entity storesdemographic information or can obtain demographic information.Deduplication is a process that is necessary to adjust cross-platformmedia exposure totals by reducing (e.g., eliminating) the doublecounting of individual audience members that were exposed to media viamore than one platform and/or are represented in more than one databaseof media impressions used to determine the reach of the media.

As used herein, a unique audience is based on audience membersdistinguishable from one another. That is, a particular audience memberexposed to particular media is measured as a single unique audiencemember regardless of how many times that audience member is exposed tothat particular media or the particular platform(s) through which theaudience member is exposed to the media. If that particular audiencemember is exposed multiple times to the same media, the multipleexposures for the particular audience member to the same media iscounted as only a single unique audience member. In this manner,impression performance for particular media is not disproportionatelyrepresented when a small subset of one or more audience members isexposed to the same media an excessively large number of times while alarger number of audience members is exposed fewer times or not at allto that same media. By tracking exposures to unique audience members, aunique audience measure may be used to determine a reach measure toidentify how many unique audience members are reached by media. In someexamples, increasing unique audience and, thus, reach, is useful foradvertisers wishing to reach a larger audience base.

An AME may want to find unique audience/deduplicate impressions acrossmultiple database proprietors, custom date ranges, custom combinationsof assets and platforms, etc. Some deduplication techniques performdeduplication across database proprietors using particular systems(e.g., Nielsen's TV Panel Audience Link). For example, suchdeduplication techniques match or probabilistically link personallyidentifiable information (PII) from each source. Such deduplicationtechniques require storing massive amounts of user data or calculatingaudience overlap for all possible combinations, neither of which aredesirable. PII data can be used to represent and/or access audiencedemographics (e.g., geographic locations, ages, genders, etc.).

In some situations, while the database proprietors may be interested incollaborating with an AME, the database proprietor may not want to sharethe PII data associated with its subscribers to maintain the privacy ofthe subscribers. One solution to the concerns for privacy is to sharesketch data that provides summary information about an underlyingdataset without revealing PII data for individuals that may be includedin the dataset. Not only does sketch data assist in protecting theprivacy of users represented by the data, sketch data also serves as amemory saving construct to represent the contents of relatively largedatabases using relatively small amounts of date. Further, not only doesthe relatively small size of sketch date offer advantages for memorycapacity but it also reduces demands on processor capacity to analyzeand/or process such data.

Sketch data may include a cardinality defining the number of individualsrepresented by the data (e.g., subscribers) while maintaining theidentity of such individuals private. The cardinality of sketch dataassociated with media exposure is a useful piece of information for anAME because it provides an indication of the number of audience membersexposed to particular media via a platform maintained by the databaseproprietor providing the sketch data. However, in some instances, sketchdata may be provided by database proprietors without providing anindication of the cardinality of the data. Even when the cardinality forsketch data is provided, problems for audience metrics arise when themedia may be accessed via multiple different database proprietors thateach provide separate sketch data summarizing the individual subscribersthat were exposed to the media. In particular, the sum of thecardinalities of each sketch data is not a reliable estimate of theunique audience size because the same individual may be represented inmultiple datasets associated with different sketch data. As a result,such individuals will be double counted (or possible more than twice ifthere are more than two datasets being aggregated) resulting in theincorrect inflation of the unique audience size. Furthermore,identifying overlap between two different sets of sketch data isnon-trivial because, as stated above, the sketch data is generated topreserve the identity and privacy of the individuals representedthereby. Examples disclosed herein overcome the above challenges byenabling the estimation of a total cardinality of users represented insketch data associated with two or more different datasets so that anAME may be able to deduplicate individuals represented in more than oneof the datasets, thereby enabling the accurate estimate of the uniqueaudience for a particular media item. Furthermore, the cardinalityestimation in examples disclosed herein may be made with or withoutdatabase proprietors providing the dataset-specific cardinalitiesassociated with the different data sketches being combined.

Notably, although third-party cookies are useful for third-partymeasurement entities in many of the above-described techniques to trackmedia accesses and to leverage demographic information from third-partydatabase proprietors, use of third-party cookies may be limited or maycease in some or all online markets. That is, use of third-party cookiesenables sharing anonymous PII subscriber information across entitieswhich can be used to identify and deduplicate audience members acrossdatabase proprietor impression data. However, to reduce or eliminate thepossibility of revealing user identities outside database proprietors bysuch anonymous data sharing across entities, some websites, internetdomains, and/or web browsers will stop (or have already stopped)supporting third-party cookies. This will make it more challenging forthird-party measurement entities to track media accesses via first-partyservers. That is, although first-party cookies will still be supportedand useful for media providers to track accesses to media via their ownfirst-party servers, neutral third parties interested in generatingneutral, unbiased audience metrics data will not have access to theimpression data collected by the first-party servers using first-partycookies. Examples disclosed herein may be implemented with or withoutthe availability of third-party cookies because, as mentioned above, thedatasets used in the deduplication process are generated and provided bydatabase proprietors, which may employ first-party cookies to trackmedia impressions from which the datasets (e.g., sketch data) isgenerated.

Although examples disclosed herein are described in association withaudience metrics related to media impressions, examples disclosed hereinmay be similarly used for other applications to deduplicate betweenmultiple different datasets while preserving privacy. The datasetsthemselves need not be audiences or email addresses. They could be, forexample, bank accounts, lists of purchased items, store visits, trafficpatterns, etc. The datasets could be represented as lists of numbers orany other information represented as unique entries in a database.

FIG. 1 shows an example environment 100 that includes an exampleaudience measurement entity (AME) 102, an example database proprietor A106 a, an example database proprietor B 106 b, and example clientdevices 108. The example AME 102 includes an example AME computer 110that implements an example audience metrics generator 112 to determineaudience sizes based on media impressions logged by the databaseproprietors 106 a-b. In the illustrated example of FIG. 1 , the AMEcomputer 110 may also implement an impression monitor system to logmedia impressions reported by the client devices 108. In the illustratedexample of FIG. 1 , the client devices 108 may be stationary or portablecomputers, handheld computing devices, smart phones, Internetappliances, smart televisions, and/or any other type of device that maybe connected to the Internet and capable of accessing and/or presentingmedia.

As used herein, an audience size is defined as a number of deduplicatedor unique audience members exposed to a media item of interest foraudience metrics analysis. A deduplicated or unique audience member isone that is counted only once as part of an audience size. Thus,regardless of whether a particular person is detected as accessing amedia item once or multiple times, that person is only counted once inthe audience size for that media item. Audience size may also bereferred to as unique audience or deduplicated audience.

As used herein, a media impression is defined as an occurrence of accessand/or exposure to media 114 (e.g., an advertisement, a movie, a movietrailer, a song, a web page banner, etc.). Examples disclosed herein maybe used to monitor for media impressions of any one or more media types(e.g., video, audio, a web page, an image, text, etc.). In examplesdisclosed herein, the media 114 may be content and/or advertisements.Examples disclosed herein are not restricted for use with any particulartype of media. On the contrary, examples disclosed herein may beimplemented in connection with tracking impressions for media of anytype or form in a network.

In the illustrated example of FIG. 1 , content providers and/oradvertisers distribute the media 114 via the Internet to users thataccess websites and/or online television services (e.g., web-based TV,Internet protocol TV (IPTV), etc.). In some examples, the media 114 isserved by media servers of the same internet domains as the databaseproprietors 106 a-b. For example, the database proprietors 106 a-binclude corresponding database proprietor servers 118 a-b that can servemedia 114 to their corresponding subscribers via the client devices 108.Examples disclosed herein can be used to generate audience metrics datathat measures audience sizes of media served by different ones of thedatabase proprietors 106 a-b. For example, the database proprietors 106a-b may use such audience metrics data to promote their online mediaserving services (e.g., ad server services, media server services, etc.)to prospective clients. By showing audience metrics data indicative ofaudience sizes drawn by corresponding ones of the database proprietors106 a-b, the database proprietors 106 a-b can sell their media servingservices to customers interested in delivering online media to users.

In some examples, the media 114 is presented via the client devices 108.When the media 114 is accessed by the client devices 108, the clientdevices 108 send impression requests 122 a-b to the database proprietorservers 118 a-b to inform the database proprietor servers 118 a-b of themedia accesses. In this manner, the database proprietor servers 118 a-bcan log media impressions in impression records of correspondingdatabase proprietor audience metrics databases 124 a-b. In someexamples, when a database proprietor server 118 a-b serves the media114, the impression request 122 a-b includes a first-party cookie set bythat database proprietor server 118 a-b so that the database proprietorserver 118 a-b can log an impression for the media 114 without using athird-party cookie. In some examples, the client devices 108 also sendimpression requests 122 c to the AME 102 so that the AME 102 can logcensus impressions in an AME audience metrics database 126. In theillustrated example of FIG. 1 , the database proprietors 106 a-b logdemographic impressions corresponding to accesses by the client devices108 to the media 114. Demographic impressions are impressions logged inassociation with demographic information collected by the databaseproprietors 106 a-b from registered subscribers of their services. Also,in the illustrated example of FIG. 1 , the AME computer 110 logscensus-level media impressions corresponding to accesses by clientdevices 108 to media 114. Census-level media impressions (e.g., censusimpressions) are impressions logged regardless of whether demographicinformation is known for those logged impressions. In some examples, thecensus impressions include some media impressions accessed via aplatform maintained by the database proprietor A 106 a and some mediaimpressions accessed via a platform maintained by the databaseproprietor B 106 b. In some examples, the AME computer 110 does notcollect impressions, and examples disclosed herein are based on audiencedata from impressions collected by the database proprietors 106 a-b. Forinstance, the AME computer 110 may not collect impressions if thedatabase proprietors 106 a-b do not allow or support third-party cookieson their platforms.

In some examples, the media 114 is encoded to include a media identifier(ID). The media ID may be any identifier or information that can be usedto identify the corresponding media 114. In some examples the media IDis an alphanumeric string or value. In some examples, the media ID is acollection of information. For example, if the media 114 is an episode,the media ID may include program name, season number, and/or episodenumber. When the example media 114 includes advertisements, suchadvertisements may be content and/or advertisements. The advertisementsmay be individual, standalone ads and/or may be part of one or more adcampaigns. In some examples, the ads of the illustrated example areencoded with identification codes (e.g., data) that identify theassociated ad campaign (e.g., campaign ID, if any), a creative type ID(e.g., identifying a Flash-based ad, a banner ad, a rich type ad, etc.),a source ID (e.g., identifying the ad publisher), and/or a placement ID(e.g., identifying the physical placement of the ad on a screen). Insome examples, advertisements tagged with the monitoring instructionsare distributed with Internet-based media content such as, for example,web pages, streaming video, streaming audio, IPTV content, etc. As notedabove, methods, apparatus, systems, and/or articles of manufacturedisclosed herein are not limited to advertisement monitoring but can beadapted to any type of content monitoring (e.g., web pages, movies,television programs, etc.).

In some examples, the media 114 of the illustrated example is tagged orencoded to include monitoring or tag instructions, which are computerexecutable monitoring instructions (e.g., Java, java script, or anyother computer language or script) that are executed by web browsersthat access the media 114 via, for example, the Internet. Execution ofthe monitoring instructions causes the web browser to send theimpression requests 122 a-c (e.g., also referred to as tag requests) toone or more specified servers of the AME 102, the database proprietor A106 a, and/or the database proprietor B 106 b. As used herein,impression requests 122 a-c are used by the client devices 108 to reportoccurrences of media impressions caused by the client devices accessingthe media 114. In the illustrated example, the impression requests 122a-b include user-identifying information that the database proprietors106 a-b can use to identify the subscriber that accessed the media 114.For example, when a subscriber of the database proprietor A 106 a logsinto a server of the database proprietor A 106 a via a client device108, the database proprietor A 106 a sets a database proprietor cookieon the client device 108 and maps that cookie to the subscriber'sidentity/account information at the database proprietor server 118 a. Inexamples disclosed herein, subscriber identity and/or subscriber accountinformation includes personally identifiable information (PII) such asfull name, street address, residence city and state, telephone number,email address, age, date of birth, social security number, demographicinformation, and/or any other personal information provided bysubscribers in exchange for services from the database proprietors 106a-b. By having such PII data mapped to database proprietor cookies, thedatabase proprietor A 106 a can subsequently identify the subscriberbased on the database proprietor cookie to determine when that useraccessed different media 114 and to log an impression in associationwith demographics and/or other PII data of that user. In the illustratedexample of FIG. 1 , the impression requests 122 a-b include databaseproprietor cookies of the client devices 108 to inform the databaseproprietors 106 a-b of the particular subscribers that accessed themedia 114. In some examples, the AME 102 also sets AME cookies in theclient devices 108 to identify users that are enrolled in a panel of theAME 102 such that the AME 102 collects PII data of people that agree tohaving their internet activities monitored by the AME 102.

The impression requests 122 a-c may be implemented using HTTP requests.However, whereas HTTP requests are network communications thattraditionally identify web pages or other resources to be downloaded,the impression requests 122 a-c of the illustrated example are networkcommunications that include audience measurement information (e.g., adcampaign identification, content identifier, and/or user identificationinformation) as their payloads. The server (e.g., the AME computer 110and/or the database proprietor servers 118 a-b) to which the impressionrequests 122 a-c are directed is programmed to log occurrences ofimpressions reported by the impression requests 122 a-c. Furtherexamples of monitoring instructions (e.g., beacon instructions) and usesthereof to collect impression data are disclosed in Mazumdar et al.,U.S. Pat. No. 8,370,489, entitled “Methods and Apparatus to DetermineImpressions using Distributed Demographic Information,” which is herebyincorporated herein by reference in its entirety.

In other examples in which the media 114 is accessed by apps on mobiledevices, tablets, computers, etc. (e.g., that do not employ cookiesand/or do not execute instructions in a web browser environment), an apppublisher (e.g., an app store) can provide a data collector in aninstall package of an app for installation at the client devices 108.When a client device 108 downloads the app and consents to theaccompanying data collector being installed at the client device 108 forpurposes of audience/media/data analytics, the data collector can detectwhen the media 114 is accessed at the client device 108 and cause theclient device 108 to send one or more of the impression requests 122 a-cto report the access to the media 114. In such examples, the datacollector can obtain user identifiers and/or device identifiers storedin the client devices 108 and send them in the impression requests 122a-c to enable the database proprietors 106 a-b and/or the AME 102 to logimpressions. Further examples of using a collector in client devices tocollect impression data are disclosed in Burbank et al., U.S. Pat. No.8,930,701, entitled “Methods and Apparatus to Collect Distributed UserInformation for Media Impressions and Search Terms,” and in Bosworth etal., U.S. Pat. No. 9,237,138, entitled “Methods and Apparatus to CollectDistributed User Information for Media Impressions and Search Terms,”both of which are hereby incorporated herein by reference in theirentireties.

In some examples, the database proprietor servers 118 a-b mayadditionally or alternatively user server logs to log impressions basedon requests for media 114 from the client devices 108. For example, whena user of a client device 108 provides a URL or selects an item of mediafor viewing, the client device 108 sends an HTTP request (e.g., theimpression request 122 a-b) to a database proprietor server 118, a-bthat includes the first-party cookie and an identifier of the requestedmedia. In response, the database proprietor server 118 a-b serves therequested media to the client device 108 and logs an impression of themedia as attributable to the client device 108.

Typically, the database(s) 124 a-b maintained by the databaseproprietors 106 a-b are implemented in a closed platform or walledgarden so that untrusted third parties do not have access to theinformation stored in the database. Among other reasons, databasesystems implemented in this manner serve to maintain the privacy of theusers registered with the database proprietors 106 a-b. Maintaining theprivacy of individuals represented within the databases of the databaseproprietors 106 a-b is in some tension with the interests of third-partyentities (e.g., media providers that may want to target particularindividuals (and/or particular demographic segments of a population)with media (e.g., advertisements), and/or the AME 102 that may want togenerate audience metrics based on tracked exposures to the media 114).

In the illustrated example, the database proprietors 106 a-b collaboratewith the AME 102 so that the AME 102 can operate as an independent partythat measures and/or verifies audience measurement informationpertaining to the media 114 accessed by the subscribers of the databaseproprietors 106 a-b. However, the database proprietors 106 a-b desire todo so while protecting the privacies of their subscribers by not sharingor revealing subscriber identities, subscriber information, and/or anyother subscriber PII data to outside parties. In examples disclosedherein, to share impression data with the AME 102 without revealingsubscriber identities, subscriber information, and/or any othersubscriber PII data, the database proprietors 106 a-b process theircollected impression data to generate corresponding sketch data 132 a-b.

As used herein, sketch data is an arrangement of data for use in massivedata analyses. For example, operations and/or queries that are specifiedwith respect to the explicit and/or very large subsets, can be processedinstead in sketch space (e.g., quickly (but approximately) from the muchsmaller sketches representing the actual data). This enables processingeach observed item of data (e.g., each logged media impression and/oraudience member) quickly in order to create a summary of the currentstate of the actual data. In some examples, summary statistics or sketchdata provide an indication of certain characteristics (e.g., number ofimpressions of a media item and/or audience reach of the media item) ofdata in a database without disclosing any personally identifiableinformation of individual users that may have contributed to the summarystatistics.

One type of data structure that is useful to provide summary statistics(e.g., sketch data) in the context of tracking exposure to media is theBloom filter array. A typical Bloom filter array is a vector or array ofbits that are initialized to 0 and then populated by flipping individualones of the bits from 0 to 1 based on the allocation or assignment ofusers (or other data entries) in a database (e.g., the databases 124 a-bof the database proprietors 106 a-b of FIG. 1 ) to respective ones ofthe bits in the Bloom filter array. The users (or other data entries) ina database that are represented in the Bloom filter array are identifiedas corresponding to summary statistics of interest (e.g., users thatwere exposed to a particular media item). That is, while it would bepossible to generate a vector for sketch data of all subscribers ofeither one of the database proprietors 106 a-b, in many instances, thesubscribers included in particular sketch data 132 a-b may be the subsetof all subscribers that corresponds to audience members that accessedand/or were exposed to a particular media item 114 of interest.

The process of generating a Bloom filter array representative of threedistinct users is demonstrated in connection with FIGS. 2-5 . FIG. 2illustrates an initial Bloom filter array 202 that has a vector lengthof 10 bits with all values being initialized to 0. FIG. 3 illustratesthe values of the elements in the Bloom filter array 202 after themapping of a first user to the Bloom filter array 202. FIG. 4illustrates the values of the elements in the Bloom filter array 202after the mapping of a second user to the Bloom filter array 202. FIG. 5illustrates the values of the elements in the Bloom filter array 202after the mapping of a third user to the Bloom filter array 202. Topopulate the Bloom filter array, email addresses 302, 402, 502 of therespective first, second, and third users are used. While the emailaddresses 302, 402, 502 are represented in the figures, any type of PIIdata could additionally or alternatively be used.

As shown in FIGS. 3-5 , three separate hash functions 304, 306, 308 areapplied to each of the email addresses 302, 402, 502 and the particularbit or element in the Bloom filter array 202 to which the correspondinguser is mapped is based on the output of the hash functions 304, 306,308. The three hash functions 304, 306, 308 are shown for purposes ofexplanation but any number of hash functions may be used (e.g., only 1hash function, 2 hash functions, more than 3 hash functions). Inexamples disclosed herein, each of the hash functions 304, 306, 308 aredesigned to map a particular input (e.g., a particular email address302, 402, 502) to one and only one element in the Bloom filter array202. Further, the hash functions 304, 306, 308 are designed such thatthe probability of a particular input being assigned to a given elementin the Bloom filter array 202 is the same as the probability of beingassigned to any other element in the Bloom filter array 202. That is,where the Bloom filter array 202 has a length of m (e.g., m=10 in theillustrated examples), the probability p_(i) that a given input (e.g., aparticular email address 302, 402, 502) is assigned to the ith elementis p_(i)=1/m.

In some examples, for the sketch data 132 a-b (e.g., the Bloom filterarray 202) from the separate database proprietors 106 a-b to be reliablyaggregated and meaningfully analyzed, the particular hash functions usedby each database proprietors 106 a-b need to be agreed upon in advance.Further, the length of the Bloom filter array 202 as generated by eachdatabase proprietors 106 a-b needs to be the same. Based on theseconstraints, if a user is a registered subscriber of both databaseproprietors 106 a-b and identified as an audience member of a particularmedia item 114, then both database proprietors 106 a-b will include theuser in their respective Bloom filter arrays (e.g., sketch data 132 a-b)and the user will be allocated to the same elements in both Bloom filterarrays (e.g., based on the same output of the same hash function used byboth database proprietors 106 a-b). Inasmuch as hashing functions cannotbe reversed, the PII data for the particular audience members is keptprivate, thereby preserving the anonymity of the underlying raw datarepresented by the sketch data 132 a-b.

As represented in FIG. 3 , the first email address 302 is allocated tothe first element of the Bloom filter array 202 based on the first hashfunction 304, the eighth element of the Bloom filter array based on thesecond hash function 306, and the fourth element of the Bloom filterarray 202 based on the third hash function. As such, the bit value ofeach of the first, fourth, and eighth elements in the Bloom filter array202 are flipped from a 0 (as shown in FIG. 2 ) to a 1 (as shown in FIG.3 ).

As represented in FIG. 4 , the second email address 402 is allocated toeach of the fourth, seventh, and eighth elements of the Bloom filterarrays 202 based on the respective outputs of the first, second, andthird hash functions 304, 306, 308. As a result, the bit vale of theseventh element in the Bloom filter array 202 is flipped from a 0 toa 1. Notably, however, there is no change in the bit values for thefourth and eighth elements in the Bloom filter array 202 because thesebits were already changed to a value 1 based on the mapping of the firstemail address 302 to the same elements. In other words, a value of 0 ina particular element in a Bloom filter array 202 remains a 0 so long asno data entry (e.g., no user) is mapped to that particular element.However, once at least one user is mapped to a particular element thevalue of the element is flipped to a 1 and remains a 1 regardless of anyfurther assignments of different users to the same element.

As represented in FIG. 5 , the third email address 502 is allocated tofifth element twice (based on each of the first and third hash functions304, 308) and to the eighth element once (based on the second hashfunction 306). As a result, the value of fifth element is flipped to a 1(based on the output of the first hash function 304) and remains a 1thereafter such that the duplicate allocation to that element (based onthe output of the third hash function 308) has no effect. Further, asabove, the allocation of the third email address 502 to the eighthelement in the Bloom filter array 202 (based on the second hash function306) has no effect on the corresponding bit value because the value waspreviously flipped to a 1.

The mapping of the output of multiple different hash functions (e.g.,the first and third hash functions 304, 308) to the same element (e.g.,the fifth element in FIG. 5 ) for a single user identifier (e.g., thirdemail address 502) is referred to as a hash collision. There is alwayssome probability that a hash collision may occur when multiple hashfunctions are used. However, the probability of a hash collision may bereduced by increasing the length of the Bloom filter array 202 (e.g.,increasing the number of elements in the array to which a user may beallocated). In many applications, the number of elements in a Bloomfilter array may number in the hundreds or even in the thousands suchthat hash collisions are relatively rare. Relatively long Bloom filterarrays also reduce the likelihood of the array becoming saturated. ABloom filter array becomes saturated when an overly large proportion ofthe bits are flipped to a value of 1. As mentioned above, once a bitvalue is flipped to a 1 in a Bloom filter array, the value remains at avalue of 1 thereafter. Thus, as the number of users to be represented ina Bloom filter array increase, there will be an ever increasing numberof 1s until (theoretically) all 0s have become 1s. When a Bloom filterarray is entirely filled with 1s (or has an overly large proportion of1s) it is no longer possible to infer anything from the data sketch.Accordingly, Bloom filter arrays are designed with a sufficient lengthrelative to an expected size of the database to be represented to reduce(e.g., avoid) saturation so that the resulting sketch data remainsmeaningful and valuable.

While longer Bloom filter arrays reduce the likelihood of hashcollisions and reduce the likelihood of saturation occurring, havingBloom filter arrays that are overly long presents concerns for userprivacy. For instance, although the Bloom filter array does not containany personally identifiable information (PII) data (e.g., the emailaddresses 302, 402, 502), the flipping of bits from 0 to 1 is based on ahash of such PII data. As such, if a Bloom filter array is sparselypopulated because of a relatively large number of elements to which eachuser may be allocated and/or a relatively small database represented inthe Bloom filter array, it is possible that separate users will bemapped to separate elements in the Bloom filter array with no overlap.In such a situation, there may be a loss of privacy if a third-partyentity has access to the Bloom filter array and has independent accessto the email addresses 302, 402, 502 and knows the particular hashfunction(s) used to populate the Bloom filter array 202. In particular,the third party may be able to confirm whether or not a particular userwas included in the sketch data represented by the Bloom filter array202 by regenerating the hashes and mapping the outputs to the Bloomfilter array 202 to see whether the corresponding elements have a bitvalue of 0 or 1. However, this privacy concern is somewhat mitigated forvery large databases and/or Bloom filter arrays with short lengthsbecause multiple user are more likely to map to the same element in theBloom filter array 202. That is, a bit value of 1 in a particularelement of the Bloom filter array 202 may correspond to multiple usersin a database the Bloom filter array 202 is created to represent suchthat a third-party entity may only confirm whether it is possible that aparticular user is included in the dataset underlying the Bloom filterarray 202. Therefore, the length of a Bloom filter array is oftendefined based on a tradeoff between increasing user privacy (by reducingthe vector length) and reducing saturation for more reliable statistics(by increasing the vector length). Notably, if a third-party entitydetermines that the output of a hash function for a particular usercorresponds to an element in the Bloom filter array 202 that has a valueof 0, the third-party entity can at least confidently confirm that theparticular user is not included in the underlying dataset. Thus, whileBloom filters can generate false positives when testing for datasetmembership, false negatives are impossible.

Even though the contents of a database may be summarized by sketch datain the form of a Bloom filter array, the mere fact of including the dataassociated with a particular user in sketch data for a correspondingdatabase still has the potential to expose the user to a loss of privacybased on differences in the summary statistics depending on whether ornot the user information of the particular user is included. Often,summary statistics shared outside of a walled garden (closed platform)system are designed to be differentially private. Summary statistics aredifferential private if a third party having access to the summarystatistics cannot determine whether the user information of a particularindividual was used in generating the summary statistics. Differentialprivacy is defined mathematically by the concept of ε-differentialprivacy, which also defines the parameters under which noise must beadded to the summary statistics to ensure the resulting summarystatistics are differentially private.

Thus, in some examples, to satisfy the requirements of differentialprivacy, noise is introduced into the Bloom filter array 202 before itis shared with other (e.g., untrusted) entities. More particularly,noise is added to the Bloom filter array 202 by flipping values ofdifferent ones of the bits in the Bloom filter array 202.

As outlined above, typical Bloom filter arrays are generated by flippingparticular elements with a value of 0 to a value of 1 after the firstassignment of a user to such elements and then retaining the value of 1regardless of how many other users are assigned to the same elements.This one direction flipping of bits from 0s to 1s can lead to saturationof the Bloom filter array. Unlike such Bloom filter arrays, examplesdisclosed herein involve the flipping of the value of a particularelement each time a user is allocated to that particular element. Thus,like traditional Bloom filter arrays, if the value of the element is 0and a user is assigned to that element, the value flips to 1. However,unlike traditional Bloom filter arrays, if the value of the element is 1(based on a previously allocated user) and another user is assigned tothat element, the value flips back to a 0. In other words, the value forany given element alternates back and forth between 0 and 1 each timeanother user is allocated to the given element. Stated differently, thefinal value for a given element in example Bloom filter arrays disclosedherein depends on whether the total number of users are assigned to thegiven element. If an even number of users are assigned to an element,the final value of the element will be the same as its initial value(e.g., the initialized 0 value will end up as a 0). By contrast, if anodd number of users are assigned to an element, the final value of theelement will be the opposite of its initial value (e.g., the initialized0 value will end up as a 1).

The final value in a Bloom filter array after all data entries (e.g.,users) have been assigned to respective elements in the Bloom filterarray may be determined based on modulo 2 arithmetic. Stated generally,in mathematics “modulo d” is defined as the remainder after dividing aninteger number by d. The possible output is any number between 0 andd−1. Two numbers are said to be congruent if they share the sameremainder. This can be stated as a≡b (mod d). For example 17 and 27 areboth congruent modulo 10 as they share the same remainder of 7 afterdividing by 10, which is written as a congruence relation as 17≡27 (mod10). The symbol for addition can be generalized in modulo arithmeticwith the symbol ⊕_(d) illustrating it is the result of the ordinaryaddition modulo d which is the final answer. We have 7⊕₁₀4=1, as 7+4=11,which, after dividing by 10, yields a remainder 1. Most people arefamiliar with modulo 12 in daily lives as ‘clock arithmetic.’

Modulo 2 arithmetic, as used in examples disclosed herein, deals withonly two numbers, {0, 1}. Applying the operation (mod 2) to any evennumber will return the value 0, whereas applying (mod 2) to an oddnumber will return the value 1. Rephrased in terms of congruentrelationships we would have (even integer)≡0 (mod 2) and (odd integer)≡1(mod 2). The full addition table for modulo 2 arithmetic is shown belowin Table 1. As can be seen in Table 1, every time we increment the valuestarting from 0 it alternates between 1 and 0 back and forth. 0⊕₂1⊕₂1=(0⊕₂1) ⊕D₂1=⊕₂1=0.

TABLE 1 Modulo 2 addition ⊕₂ 0 1 0 0 1 1 1 0

Alternating between 0 and 1 based on modulo 2 addition for increasingvalues is equivalent to flipping the value between 0 and 1 at everyassignment of the Bloom filter as demonstrated in connection with FIGS.6-9 . For purposes of clarity, a Bloom filter array generated byrepeatedly flipping the values of elements back and forth between 0 and1 as disclosed herein is referred to as a modulo 2 Bloom filter array todistinguish it from a traditional Bloom filter array in which once anelement is flipped to a value of 1 it remains at a value of 1. That is,as used herein, the term “Bloom filter array” is a generic term thatincludes both traditional Bloom filter arrays and modulo 2 Bloom filterarrays. FIG. 6 illustrates an initial modulo 2 Bloom filter array 602that has a vector length of 10 bits with all values being initialized to0 in a similar manner to the traditional Bloom filter array 202 of FIG.2 . FIG. 7 illustrates the values of the elements in the modulo 2 Bloomfilter array 602 after the mapping of a first user to the Bloom filterarray 602. FIG. 8 illustrates the values of the elements in the modulo 2Bloom filter array 602 after the mapping of a second user to the Bloomfilter array 602. FIG. 9 illustrates the values of the elements in themodulo 2 Bloom filter array 602 after the mapping of a third user to theBloom filter array 602. In the illustrated example of FIGS. 6-9 , thesame three hash functions as used in FIGS. 2-5 are applied to the sameemail addresses 302, 402, 502 as in FIGS. 2-5 . As a result, theparticular elements to which each user is assigned in the modulo 2 Bloomfilter array 602 of FIGS. 6-9 correspond to the same elements to whichthe users were assigned in the traditional Bloom filter array 202 ofFIGS. 2-5 .

Thus, after the first user has been assigned to the modulo 2 Bloomfilter array 602 as represented in FIG. 7 , the first, fourth, andeighth elements are flipped to a value of 1 in a similar manner to thetraditional Bloom filter array 202 in FIG. 3 . However, after assignmentof the second user to the fourth, seventh, and eighth elements, as shownin FIG. 8 , the modulo 2 Bloom filter array 602 differs from thetraditional Bloom filter array 202 at the corresponding pointrepresented in FIG. 4 . More particularly, rather than the fourthelement remaining at a value of 1 as in FIG. 4 , the bit value isflipped back to a 0 in FIG. 8 because of a second entry assigned to thefourth element. Similarly, the eighth element in the modulo 2 Bloomfilter array 602 is flipped back to a 0 in FIG. 8 (from a 1 previouslyin FIG. 7 ) because of a second entry assigned to the eighth element.The seventh element in the modulo 2 Bloom filter array 602 of FIG. 8 waspreviously a 0 (FIG. 7 ) and, therefore, is flipped to a 1 in a similarmanner to the traditional Bloom filter array 602 of FIG. 4 . In FIG. 9 ,the fifth element is flipped to a 1 based on the output of the firsthash function but then flipped back to a 0 based on the output of thesecond hash function. That is, the value of the fifth element remainsthe same as before because two entries (e.g., an even number of entries)are assigned to the same element. Further, eighth element in the modulo2 Bloom filter array 602 is flipped back to a 1 for a second time inFIG. 9 because a third entry is assigned to the eighth element.

In the illustrated example of FIG. 9 , every element in the Bloom filterarray 602 that is associated with an odd number of hash function outputsmapped thereto has a value of 1. That is, the first and seventh elementshave a value of one because only a single user was assigned to them froma single hash function. Further, the eighth element also has a value of1 because each of the three separate users were assigned to the eighthelement based on a single hash function each. Every element in the Bloomfilter array 602 of FIG. 9 that is associated with an even number(including 0) of hash function outputs mapped thereto has a value of 0.For instance, the second, third, sixth, ninth, and tenth elements areall 0 because no users were assigned to those elements. Further, thefourth and fifth elements have a value of 0 because each element wasassigned a user twice. More particularly, the fourth element wasassigned the first and second user once each and the fifth element wasassigned the third user twice (based on two different hash functions).

As can be seen, the modulo 2 Bloom filter array 602 of FIG. 9 includesfewer is in the array than the traditional Bloom filter array 202 ofFIG. 5 because the fourth and fifth elements have a value of 0 in FIG. 9, whereas the same elements have a value of 1 in FIG. 5 . The fewerelements with a value of 1 resulting from the modulo 2 approach togenerating the Bloom filter array 602 represented in FIGS. 6-9 eliminateconcerns of saturation (e.g., an overly large proportion of the elementsin the Bloom filter array becoming a 1). Indeed, regardless of how manyentries are in a database, it is highly unlikely that the Bloom filterarray 602 will ever become all is (or substantially all 1s) becauseevery other assignment of a new entry to any particular element willflip the element back to the value of 0. By contrast, the traditionalBloom filter array generation process outlined above in connection withFIGS. 2-5 will ultimately result in the Bloom filter array 202 reachingsaturation with all or substantially all elements having a value of 1.As a result, the modulo 2 approach to Bloom filter array generationdisclosed herein provides a technical advantage over the traditionalapproach because the same information can be represented in a Bloomfilter array of a shorter length because there is less concern of thevector saturating with all 1s. A shorter Bloom filter array provides atechnical advantage because it takes up less memory space and can beanalyzed with greater efficiency (e.g., less processing capacity and/orin less time) than a traditional Bloom filter array that requires alonger length to avoid concerns of saturation.

The modulo 2 approach to generating Bloom filter arrays disclosed hereinalso provides for increased privacy because it eliminates thepossibility of either confirming the presence or absence of a particularuser within an underlying dataset. That is, whereas traditional Bloomfilters make false negatives for testing the membership of a user in adataset impossible such that a user can be conclusively confirmed to notbe in the dataset, a user cannot be conclusively confirmed to be eitherincluded in the dataset or excluded from a dataset from a modulo 2 Bloomfilter. Furthermore, this level of privacy is achieved without the needfor adding noise. As a result, examples disclosed herein further save onprocessing capacity by eliminating additional operations associated withthe adding of noise to Bloom filter arrays before they may be sharedwith third-party entities. Of course, while adding noise is no longernecessary, in some examples, noise may nevertheless still be added tofurther increase the level of privacy protection offered by the Bloomfilter arrays disclosed herein.

The improved privacy achieved by example modulo 2 Bloom filter arraysdisclosed herein may be demonstrated with reference to the allocationsof 5 users to two elements (e.g., using two hash functions) of a lengthm=5 element array.

TABLE 2 Allocations of Users across 5 element Bloom Filter Array usingTwo different hash Functions Bit array element allocation Bit arrayelement allocation User Name based on Hash 1 based on Hash 2 Alice 2 4Bob 1 3 Carol 2 4 Dave 1 2 Eve 4 5

As represented in Table 2, Alice is assigned to the second and fourthelements of the five-element Bloom filter array. Thus, testing whetherAlice is included in a traditional Bloom filter array representing anunderlying dataset corresponding to a subset of the five users listed inTable 2 requires confirming that both the second and fourth elements inthe Bloom filter array have a value of 1. However, this cannotconclusively confirm that Alice is in the dataset represented in theBloom filter array; only that it is possible that Alice may be in thedataset. The reason for this uncertainty is that the values of 1 in thesecond and fourth elements may be attributed to other users that wereassigned to the same elements. In particular, as shown in Table 2, Carolis also assigned to the second and fourth elements such that there is noway of knowing for certain whether Alice is included in the dataset andto claim she is included when it is really Carol would be a falsepositive. Another possible scenario giving rise to a false positive forthe inclusion of Alice would be a dataset that includes only Dave andEve. Dave is assigned to the second element and Eve is assigned to thefourth element, so Dave and Eve collectively result in both elementsassociated with Alice being flipped to a value of 1. By contrast, if thedataset underlying a traditional Bloom filter array included only Boband Dave, only the first three elements in the Bloom filter array wouldbe flipped to the value of 1. As a result, the fourth element would be a0 and it could be conclusively determined that Alice is not in thedataset.

Unlike the traditional Bloom filter array, neither membership nornon-membership of a particular user (e.g., Alice) in an underlyingdataset associated with a modulo 2 Bloom filter array can be conclusivetested or determined. That is, neither the presence nor the absence ofthe user in the dataset is definitive or guaranteed. The uncertainty isachieved by the repeated bit flipping between 0 and 1 as multiple usersare assigned to the same elements in the Bloom filter array. Table 3identifies two example datasets (and the resulting array of values in acorresponding modulo 2 Bloom filter array) based on different subsets ofusers selected from the full set of five users shown in Table 2 for eachof four different scenarios including: (I) Alice is in the dataset andboth assigned elements associated with Alice have a value of 1; (II)Alice is not in the dataset but both assigned elements associated withAlice have a value of 1 (this is the false positive scenario for atraditional Bloom filter array noted above); (III) Alice is in thedataset but the assigned elements associated with Alice are not both 1;and (IV) Alice is not in the dataset and the assigned elementsassociated with Alice are not both 1.

TABLE 3 Example Datasets and Associated Bloom filter Arrays Alice indataset Alice not in dataset Assigned Ia) {Alice} IIa) {Carol} elementsfor (01010) (01010) Alice both Ib) {Alice, Bob} IIb) {Dave, Eve} equalto 1 (11110) (11011) Assigned IIIa) {Alice, Carol} IVa ) {Bob} elementsfor (00000) (10100) Alice not IIIa) {Alice, Dave, Eve} IVb) {Carol,Dave, Eve} both equal (10001) (10001) to 1

As can be seen with reference to Table 3, the assigned elementsassociated with Alice (e.g., the second and fourth elements) having avalue of 0 does not necessarily mean that Alice is not in the dataset.Rather, the values of 0 only means that an even number of assignmentshave been made to the assigned elements associated with Alice. Forinstance, the dataset IIIa includes Alice but the second and fourthelements are nevertheless both 0 because the assignment of Carol (alsoin the dataset) to the same elements cancels or reverses the bitflipping that would have resulted from the assignments associated withAlice. That is, there are an even number (e.g., two) assignments to boththe second and fourth elements such that the final value of the elementsends up at 0.

Notably, all four of the example datasets in the bottom row of Table 3include an exact mismatch with the assigned elements for Alice (e.g.,both the second and fourth elements are 0 rather than 1). However, theremay be other combinations where either the second element or the fourthelement ends up a 0 while the other element is a 1. Such element valuesin a traditional Bloom filter array would conclusively establish thatAlice is not included in the underlying dataset. However, the sameconclusion cannot be made when such element values are in a modulo 2Bloom filter array. Rather, any combination of 0s and/or 1s is possiblewhether or not Alice is included in the underlying dataset. Eachcombination may have a different probability of likelihood that Alice isincluded in the dataset, but none would equal 0% or 100%. Thus, bothfalse positives and false negatives are always possible when testing formembership of a particular user within a dataset represented by a modulo2 Bloom filter array.

While the property of individual inferential information on test entriesis unavailable for modulo 2 Bloom filter arrays, the cardinality ortotal number of unique entries in the underlying dataset maynevertheless still be estimated. Furthermore, cardinality estimationsmay be made across multiple modulo 2 Bloom filter arrays. Themathematical principles underlying the ability to estimate thecardinality of such Bloom filter arrays is the same principlecorresponding to the classic problem in probability theory involving theflipping of a biased coin. Specifically, assuming there is a biased coinwith the probability of getting a head being p and tail being q=1−p, theproblem is to then determine what the probability is to get an evennumber of heads after n tosses of the coin. The solution to this problemis expressed below in Equation 1. The probability of getting an oddnumber of heads is expressed in Equation 2.Pr(#H is even)=½(1+(q−

)^(n))=½(1+(1−2

)^(n))  Eq. 1Pr(#H is odd)=½(1−(q−

)^(n))=½(1−(1−2

)^(n))  Eq. 2

Using the probabilities defined in Equations 1 and 2 can be used toestimate the cardinality of a modulo 2 Bloom filter array, B, of lengthm with initial values B_(i)=0 for i={1, 2, . . . , m}. Where B_(i) isinitially set to 0, it will remain 0 only if an even number ofassignments were made to element or index i out of the n possibletrials. Where each index is equally likely, the probability of thatspecific element or position in the Bloom filter array being picked isp=1/m, which is equivalent to the bias of the coin used in the theoremassociated with Equations 1 and 2 above. Thus, if B_(i) is initially setto 0, the probability that the final output after n assignments thatB_(i) is still 0 is identical to observing an even number of heads aftern tosses of a biased coin where the probability of seeing a head forthat coin is p=1/m.

The exact likelihood of a particular element in a Bloom filter arraybeing associated with an even number of allocations has to consider thefull joint distribution across all 2^(m) combinations of possibleoutcomes after doing exactly n allocations. However, if n is largeenough (relative to the array length m) the likelihood can beapproximated by assuming independence across the m elements of the Bloomfilter array.

Let c_(E) be the count of elements in the Bloom filter array that wereassigned an even number of entries and c_(O) be the count of elements inthe Bloom filter array that were assigned an even number of entries.Where the initial values of the Bloom filter array are all 0, c_(E)corresponds to the number of elements with a value of 0 after allassignments or allocations have been made and c_(O) corresponds to thenumber of elements with a value of 1 after all assignments orallocations have been made. As each element in the Bloom filter array isassigned either an even number of times (to end up with a value of 0) oran odd number of times (to end up with a value of 1), the sum of c_(E)and c_(O) equals the total number of elements in the Bloom filter array(e.g., c_(E)+c_(O)=m). Assuming independence across the elements (e.g.,based on a large n), the likelihood of obtaining counts of {c_(E),c_(O)}, is the binomial probability distribution.

$\begin{matrix}{{\left( n \middle| \left\{ {c_{E},c_{O},p_{E}} \right\} \right)} = {\begin{pmatrix}m \\c_{E}\end{pmatrix}{p_{E}^{c_{E}}\left( {1 - p_{E}} \right)}^{c_{O}}}} & {{Eq}.\mspace{14mu} 3}\end{matrix}$

where

_(E) is the probability of getting an even number of heads after ntosses of a biased coin (as expressed in Equation 1) with a probabilityof getting heads being p=1/m). Thus, substituting Equation 1 intoEquation 3 yields.

$\begin{matrix}{{\left( n \middle| \left\{ {c_{E},c_{O},p} \right\} \right)} = {\begin{pmatrix}m \\c_{E}\end{pmatrix}\left( {\frac{1}{2}\left( {1 + \left( {1 - {2p}} \right)^{n}} \right)} \right)^{c_{E}}\left( {\frac{1}{2}\left( {1 - \left( {1 - {2p}} \right)^{n}} \right)} \right)^{c_{O}}}} & {{Eq}.\mspace{14mu} 4}\end{matrix}$

Taking the logarithm and derivative of Equation 4 with respect to nyields that the maximum likelihood occurs when

$\begin{matrix}{{\frac{{c_{E}\left( {1 - {2p}} \right)}^{n}{\log\left( {1 - {2p}} \right)}}{1 + \left( {1 - {2p}} \right)^{n}} - \frac{{c_{O}\left( {1 - {2p}} \right)}^{n}{\log\left( {1 - {2p}} \right)}}{1 - \left( {1 - {2p}} \right)^{n}}} = 0} & {{Eq}.\mspace{14mu} 5}\end{matrix}$Solving for n in Equation 5 yields

$\begin{matrix}{\overset{\hat{}}{n} = {\frac{\ln\left( \frac{c_{E} - c_{O}}{c_{E} + c_{O}} \right)}{\ln\left( {1 - {2p}} \right)} = \frac{\ln\left( \frac{c_{E} - c_{O}}{m} \right)}{\ln\left( {1 - {2p}} \right)}}} & {{Eq}.\mspace{11mu} 6}\end{matrix}$

where the notation of {circumflex over (n)} indicates that Equation 6 isan estimate (based on the maximum likelihood) of the number ofassignments to the Bloom filter array.

As mentioned above, each entry in a dataset represented in a Bloomfilter array may be assigned to the array multiple times based onmultiple different hash functions. Thus, in some situations the numberof assignments to the Bloom filter array is not necessarily thecardinality of the dataset but the cardinality multiplied by k hashfunctions used to assign entries in the dataset to the Bloom filterarray. Thus, for the estimate of {circumflex over (n)} to reflect theestimate of the cardinality of the Bloom filter array, Equation 6 needsto be divided by k as shown in Equation 7.

$\begin{matrix}{\overset{\hat{}}{n} = {\left( \frac{1}{k} \right)\frac{\ln\left( \frac{c_{E} - c_{O}}{m} \right)}{\ln\left( {1 - {2p}} \right)}}} & {{Eq}.\mspace{14mu} 7}\end{matrix}$

As long as c_(E)≠c_(O), the argument inside the logarithm in Equation 7is not zero such that an estimate for the cardinality (e.g., {circumflexover (n)}) is obtainable. Equality between the counts (e.g., c_(E) andc_(O)) can only occur when exactly half of the elements in the Bloomfilter array were assigned an even number of entries (resulting in afinal value of 0) and exactly half the elements were assigned an oddnumber of entries (resulting in a final value of 1). For convenience ofnotation, c_(E) may be redefined as the count c₀ of the number of 0s inthe Bloom filter array and c_(O) may be redefined as the count c₁ of thenumber of is in the Bloom filter array. The counts c₀ and c₁ can only beequal (albeit with a relatively small probability) when the length m ofthe Bloom filter array (e.g., the total number of elements in the array)is even. Accordingly, in some examples, the length of the array isdefined to be odd, thereby eliminating the possibility of having anequality between the counts of 0s and 1s in the Bloom filter array.

While an odd length m for a Bloom filter array avoids the possibilitythe argument of the logarithm in Equation 7 from equally zero, there isthe possibility that c_(E)<c_(O) (e.g., c₀<c₁) resulting in the argumentbeing negative. To avoid a negative argument in the logarithm, Equation7 may be revised by taking the absolute value of the difference betweenthe counts (with c₀ replacing c_(E) and c₁ replacing c_(O)) as follows:

$\begin{matrix}{\overset{\hat{}}{n} = {\left( \frac{1}{k} \right)\frac{\ln\left( \frac{{c_{0} - c_{1}}}{m} \right)}{\ln\left( {1 - {2p}} \right)}}} & {{Eq}.\mspace{14mu} 8}\end{matrix}$Revising Equation 7 as shown in Equation 8 is appropriate because ofsymmetry between a Bloom filter array initialized to all 0s and a Bloomfilter array initialized to all 1s. That is, if a first Bloom filterarray initially beginning with all 0s ends up with a greater number of1s than 0s after all assignments have been made, a second Bloom filterarray initially beginning with all 1s will end up with a greater numberof 0s than 1s after the same assignments have been made. Furthermore,the number of 1s and 0s in the first Bloom filter array will correspondto the respective number of 0s and 1s in the second Bloom filter array.

Typically, counting the number of 1s in an array is easier from aprocessing standpoint (based on simple addition of the bit values), theexpression of cardinality can be rephrased in terms of c₁ itself basedon the definition that c₀+c₁=m to yield

$\begin{matrix}{\overset{\hat{}}{n} = {\left( \frac{1}{k} \right)\frac{\ln\left( {{1 - \frac{2c_{1}}{m}}} \right)}{\ln\left( {1 - {2p}} \right)}}} & {{Eq}.\mspace{14mu} 9}\end{matrix}$

As a specific example, consider a scenario where a total of n=2000 items(e.g., users in a database) are allocated to individual elements of amodulo 2 Bloom filter array having a length m=1001 using a single hashfunction for a single allocation of each item (e.g., k=1). The length ofthe array is odd to eliminate the possibility of equality in the numberof 0s and 1s in the array. In this example, the output Bloom filterarray includes an array of bits in which a total of 510 elements had avalue of 1 (e.g., c₁=510). In this example, the true value for n isunknown but corresponds to the cardinality of the Bloom filter array tobe estimated. With the probability of p=1/m, the cardinality of theBloom filter array may be estimated by evaluating Equation 9, whichresults in an estimate of {circumflex over (n)}=1982.16.

Equation 9 defines the estimate for the cardinality of a single modulo 2Bloom filter array. However, in some situations, multiple differentBloom filter arrays may be provided from different entities. Forinstance, in some examples, each of the database proprietors 106 a-b ofFIG. 1 may provide a separate modulo 2 Bloom filter array (e.g., thesketch data 132 a-b) to the AME 102 for aggregation and analysis. Insome examples, the same media may be provided to audience memberscorresponding to users (e.g., subscribers) of both the databaseproprietors 106 a-b. Accordingly, in some such examples, both databaseproprietors 106 a-b may generate a corresponding Bloom filter arrayrepresenting summary statistics of the registered users (e.g.,subscribers) of each database proprietor 106 a-b that were exposed tothe media. Based on the Bloom filter arrays obtained from each of thedatabase proprietors 106 a-b, the AME 102 may estimate the total numberof unique (e.g., deduplicated) individuals that were exposed to themedia. That is, the AME 102 may estimate the reach of the media. Achallenge in making this determination is that some users registeredwith the first database proprietor 106 a may also be registered with thesecond database proprietor 106 b. If such users are exposed to the samemedia via both database proprietors 106 a-b, both database proprietorswould separately report the users' exposure to the media in theirrespective Bloom filter arrays resulting in a duplicate reporting of theuser as an audience member exposed to the media. Furthermore, asdescribed above, the summary statistics contained in the Bloom filterarrays are differentially private such that the AME 102 cannot directlyconfirm whether a user is included in one, both, or neither Bloom filterarray to appropriately resolve the duplication of audience membersacross different filters.

As outlined above, the cardinality for each modulo 2 Bloom filter array(e.g., the sketch data 132 a-b) provided from each database proprietor106 a-b may be estimated. However, the cardinality of the union of thesketch data 132 a-b from both database proprietor 106 a-b cannot bedirectly determined from these separate cardinality estimates becauseone or more of the subscribers represented in the sketch data A 132 amay also be represented in the sketch data B 132 b. Examples disclosedherein enable the deduplication of audience members across both datasetsto estimate the true unique audience for the particular media ofinterest. Furthermore, examples disclosed herein may be used to estimatethe cardinality across more than two datasets when all of the datasetsare represents by a corresponding modulo 2 Bloom filter array asdiscussed above. That is, the Bloom filter array generated by eachdatabase proprietor will have the same length m and the entries in theirrespective databases will be allocated to individual elements in theBloom filter array based on the same hash function(s). In other words,the allocation of the user “John Smith” will be to the same element inevery Bloom filter array associated with each database proprietor forwhich “John Smith” is included in the underlying dataset represented bythe corresponding Bloom filter array.

Assume that there are two database proprietors 106 a-b, each of whichgenerates respective bit arrays of length m, {B⁽¹⁾, B⁽²⁾} based on themodulo 2 addition methodology outlined above and based on the same setof hash functions. For purposes of explanation, assume that the lengthof the Bloom filter arrays is m=9 and that the values in the two Bloomfilter arrays generated by the two database proprietors 106 a-b are asfollows:B ⁽¹⁾={1,1,0,0,1,0,0,1,1}  Eq. 10B ⁽²⁾={1,0,0,1,0,1,0,1,0}  Eq. 11

The bit-wise modulo 2 addition between the two arrays is a new array asshown below:

$\begin{matrix}\frac{\begin{matrix}{\mspace{34mu}\left\{ {1,1,0,0,1,0,0,1,1} \right\}} \\{\oplus \left\{ {1,0,0,1,0,1,0,1,0} \right\}}\end{matrix}}{\mspace{31mu}\left\{ {0,1,0,1,1,1,0,0,1} \right\}} & {{Eq}.\mspace{14mu} 12}\end{matrix}$The output array shown in Equation 12 is equivalent of doing bit-wiseB_(i) ⁽¹⁾ ⊕₂ B_(i) ⁽²⁾ for each index i={1, 2, . . . , m}. Despite beinga bit array of length m derived from a union between two other arrays,the resulting array is not the modulo 2 Bloom filter array of the unionof the two underlying datasets. This is because for the allocation ofany user that belongs to both datasets is the same for both originalBloom filter arrays resulting in an even number of identicalassignments. As noted above, an even number of assignments in modulo 2arithmetic is identical to zero. Thus, a user that is represented in theBloom filter arrays provided by both database proprietors 106 a-b iscounted twice in the bit-wise union, which has the effect of leaving thevalue unchanged. In other words, users included in both underlyingdatasets effectively become invisible during the bit-wise modulo 2addition shown above. Graphically, this can be illustrated as a Venndiagram of exclusive-or across two sets.

Let the variable X represent the exclusive-or cardinality of the unionof modulo 2 Bloom filter arrays (e.g., the total number of users ineither the first Bloom filter array or the second Bloom filter array butnot both arrays). The exclusive-or cardinality X is distinct from thetrue cardinality N of the union of Bloom filter arrays (e.g., the totaldeduplicated number of users across both Bloom filter arrays regardlessof whether the users are in one or both). Further, let the variable{circumflex over (X)} represent the estimate of the true value of X asthe output of Equation 9. That is, {circumflex over (X)} is the same as{circumflex over (n)}, but the different notation is now used becausemultiple Bloom filter arrays are now involved. To use the above example,let {circumflex over (X)}_({1}), {circumflex over (X)}_({2}), and{circumflex over (X)}_({1,2}) represent the respective estimates foreach of the three arrays shown in Equation 12.

$\begin{matrix}{\mspace{45mu}{\left. \left\{ {1,1,0,0,1,0,0,1,1} \right\}\rightarrow{\overset{\hat{}}{X}}_{\{ 1\}} \right.\frac{\left. {\oplus \left\{ {1,0,0,1,0,1,0,1,0} \right\}}\rightarrow{\overset{\hat{}}{X}}_{\{ 2\}} \right.}{\mspace{40mu}\left. \left\{ {0,1,0,1,1,1,0,0,1} \right\}\rightarrow{\overset{\hat{}}{X}}_{\{{12}\}} \right.}}} & {{Eq}.\mspace{14mu} 13}\end{matrix}$

Let n₁₀ be the number of unique users in the first dataset but not inthe second dataset, n₀₁ be the number of unique users in the seconddataset but not in the first dataset, and n₁₁ be the number of uniqueusers in both datasets. These variables are referred to herein as thedisjoint cardinalities for the union of two datasets because they form acollection of mutually exclusive and exhaustive sets across allpossibilities that memberships may occur between the datasets and theirusers included therein. Based on properties of exclusive-or unions,equalities between the disjoint cardinalities and the true exclusive-orcardinalities may be expressed as follows:n ₁₀ +n ₁₁ =X _({1})n ₀₁ +n ₁₁ =X _({2})n ₁₀ +n ₀₁ =X _({1,2})  Eq. 14

If ordinary addition across the expressions in Equation 14 areperformed, each disjoint cardinality is added exactly twice.

$\begin{matrix}\frac{\begin{matrix}\; & {{n_{10} + n_{11}} = X_{\{ 1\}}} \\\; & {{n_{01} + n_{11}} = X_{\{ 2\rbrack}} \\ + & {{n_{10} + n_{01}} = X_{\{{1,2}\}}}\end{matrix}}{{2\left( {n_{01} + n_{10} + n_{11}} \right)} = {X_{\{ 1\}} + X_{\{ 2\}} + X_{\{{1,2}\}}}} & {{Eq}.\mspace{14mu} 15}\end{matrix}$

As n₀₁+n₁₀+n₁₁ is the total number N of unique users across bothdatasets, Equation 15 can be expressed in terms of N as2N=X _({1}) +X _({2}) +X _({1,2}) Eq. 16Inasmuch as each exclusive-or cardinality may be estimated usingEquation 9, as outlined above, the estimate of the total cardinality ofthe union of two modulo 2 Bloom filter arrays may be determined bydividing the sum of those estimates by two.{circumflex over (N)}=½({circumflex over (X)} _({1}) +{circumflex over(X)} _({2}) {circumflex over (X)} _({1,2}))  Eq. 17Equation 17 is true as an estimate of {circumflex over (N)} regardlessof the number of hash functions used to allocate each user to the Bloomfilter arrays (e.g. regardless of k) because each exclusive-orcardinality estimate within the sum has already taken into account thatmultiplicity of k via the

$\left( \frac{1}{k} \right)$factor in Equation 9.

While Equation 17 defines the estimate for the cardinality of uniqueusers across two modulo 2 Bloom filter arrays, the above methodology maybe generalized to any number of Bloom filter arrays. For purposes ofdiscussion, let r represent the number of different Bloom filter arrays,{B⁽¹⁾, . . . , B^((r))}, to be combined. By definition of disjointcardinalities, their sum must equal the total cardinality, as shownbelow for the number of Bloom filter arrays r being 1, 2, or 3.N=n _(i) (r=1)N=n ₀₁ +n ₁₀ +n ₁₁ (r=2)N=n ₀₀₁ +n ₀₁₀ +n ₀₁₁ +n ₁₀₀ +n ₁₀₁ +n ₁₁₀ +n ₁₁₁ (r=3)  Eq. 18

As bit-wise modulo 2 addition among a subset of the modulo 2 Bloomfilter arrays is equivalent to the same modulo 2 procedure on theexclusive-or union of their respective set memberships, any individualone of the arrays or any two or more of the r arrays, up to all r arraystaken together, may be analyzed. These different combinations of thearrays provide 2^(r)−1 estimates of the cardinality of variousexclusive-or set unions. Thus, if there are three database proprietors106 a-b each providing a separate Bloom filter array, the resultingarray after doing modulo 2 addition across all three arrays wouldproduce an estimate of the true value X_({1,2,3}). As above, this is notthe estimate of the total number of users across the union of all threedatasets, but is the estimate of the number of users in either only onedataset or all three datasets together. Those users that are in any twodatasets (but not the third) would effectively cancel because beingallocated identically in two Bloom filter arrays (e.g., an even numberof times) effectively erases the previous allocation after the modulo 2addition. This is true in general for any combination of Bloom filterarrays.

As a further illustration, consider an output array based on theexclusive-or union of a possible subset of different Bloom filter arraysbeing {B⁽¹⁾, B⁽⁴⁾, B⁽⁵⁾, B⁽⁸⁾, B⁽⁹⁾}. The output of using Equation 9 onthe bit-wise modulo 2 addition of these five arrays would produce anestimate of true exclusive-or cardinality X_({1,4,5,8,9}). This wouldestimate the total number of users in any odd-numbered combination ofthose datasets. In other words, this estimation would represent thetotal number of users that are included in only a single dataset (e.g.,{1}, {4}, {5}, {8}, or {9}), included in any combination of threedatasets (e.g., {1, 4, 5}, {1, 4, 8}, etc.), and included in all 5datasets together. Those users in an even-numbered combination of thedatasets would not be included in the estimate of X_({)1, 4, 5, 8, 9}because they self-cancel after modulo 2 addition.

As noted above in Equation 15, for two Bloom filter arrays, eachdisjoint cardinality (e.g., n₀₁, n₁₀, n₁₁) appears exactly twice in thefinal summation of all exclusive-or cardinality estimations. When the2^(r)−1 combinations of bit-wise modulo 2 addition for r Bloom filterarrays are expanded out and summed in a similar manner to Equation 15,it can be shown that each disjoint cardinality appears exactly 2^((r-1))times. In particular, let the disjoint cardinality under interest have rindices in the subscript, indicating either {0, 1} for Boolean false ortrue, respectively, if the users are included in the jth dataset, withj={1, . . . , r}. Assume that s number of those indices are true withr−s indices being false (e.g., n₁₀₁ would have r=3 and s=2, where theusers are in the 1st and 3rd datasets but not in the 2nd). Including theempty set along with all other possible combinations of the r indicesbeing chosen any of {0, 1, 2, . . . , r} at a time results in a total of2^(r) combinations. This collection is equivalent to first selecting anysubset of true indices (2^(s) combinations) and then independentlyselecting any subset of false indices (2^((r-s)) combinations). Thetotal number of combinations is still 2^(r) as we have2^(r)=2^(s)×2^((r-s)). Within the 2^(s) ways of selecting the trueindices, exactly half will have even parity and half will have oddparity. As only odd parity combinations are included within theexclusive-or cardinality expansion, the total number of times thatdisjoint cardinality under interest appears across all 2^(r)exclusive-or combinations is therefore 2^((r-s))(2^(s)/2)=2^((r-1)).This is independent of s and, therefore, valid for any disjointcardinality. Additionally, as the empty set yielded an even parity itdid not impact the number of odd parity combinations. As such, everydisjoint cardinality would appear exactly 2^((r-1)) times across all theexpansion of exclusive-or set combinations, even if excluding the emptyset within the combination.

As the sum of all disjoint cardinalities is the total cardinality ofunique users across the datasets, and each disjoint cardinality iscounted the same number of times, we can bring the constant out of thesum to provide the following general expression

$\begin{matrix}{{2^{({r - 1})}N} = {\sum\limits_{i\;\epsilon\;\Omega}X_{\{ i\}}}} & {{Eq}.\mspace{14mu} 19}\end{matrix}$where Ω is the enumeration of all combinations of subsets of {1, 2, . .. , r} taken 1 at a time, 2 at a time, etc. up to r at a time. The righthand side of Equation 19 is a sum across all 2^(r)−1 differentexclusive-or cardinalities. By replacing the true exclusive-orcardinality, X, with its respective estimate {circumflex over (X)}(corresponding to {circumflex over (n)} in Equation 9), and dividing bythe multiplicative constant, results in an expression for the estimationof the total unique cardinality across the union of all datasets:

$\begin{matrix}{\hat{N} = {2^{({1 - r})}{\sum\limits_{i\;{\epsilon\Omega}}{\hat{X}}_{\{ i\}}}}} & {{Eq}.\mspace{14mu} 20}\end{matrix}$As with Equation 17, Equation 20 is true regardless of the number ofhash functions used to allocate users to the respective Bloom filterarrays because the number of hash functions k is taken into account inEquation 9. Notably, in addition to estimate the total cardinalityacross all datasets, it is possible to estimate individual datasetintersections and, by extension, any Boolean operation of datasetmemberships by using the duality of the inclusion-exclusion principle.

For purposes of explanation, an example using actual numbers for theunion of three different datasets is provided below. Notably, thefollowing example, uses example datasets with relatively smallcardinalities that are represented in modulo 2 Bloom filter arrays ofrelatively short length. In many applications, the Bloom filter arraysmay significantly longer lengths (e.g., elements numbering in the 1000s)with values representative of underlying datasets having significantlylarger cardinalities (e.g., millions or more). With that stated, thefollowing example includes the disjoint cardinalities across threedatasets shown below:n ₀₀₁=13n ₀₀₁=10n ₀₁₁=4n ₁₀₀=11n ₁₀₁=10n ₁₁₀=17n ₁₁₁=10  Eq. 21yielding a total of 75 unique individuals. Notably, the individualdisjoint cardinalities in Equation 21 and the resulting totalcardinality would not be known in an actual scenario but are the valuesto be estimated.

The total number of individuals represented in each respective datasetin this example isX _({1}) =n ₁₀₀ +n ₁₀₁ +n ₁₁₀ +n ₁₁₁=48X _({2}) =n ₀₁₀ +n ₀₁₁ +n ₁₁₀ +n ₁₁₁=41X _({3}) =n ₀₀₁ +n ₀₁₁ +n ₁₀₁ +n ₁₁₁=37  Eq. 22Notably, Equation 22 uses the variable X defined above as theexclusive-or cardinality. Using this notation to represent the totalcardinality for a single dataset is appropriate because the exclusive-orcardinality for a single dataset is the cardinality of the set itself.In some examples, the cardinalities of each individual dataset (as shownin Equation 22) may be unknown, but are provided here for purposes ofexplanation. However, in some examples, the sketch data 132 a-b providedby the database proprietors 106 a-b may include both the Bloom filterarray and the cardinality of the underlying dataset such that the valuesin Equation 22 may be known.

In this example, all three database proprietors agreed on k=3 differenthash functions and a Bloom filter array length of m=101. The finalvalues for all elements in each of the modulo 2 Bloom filter arraysgenerated by each of the database proprietors is shown below in full

$\begin{matrix}{{{B^{(1)} = \begin{matrix}\left\{ {1,} \right. & {0,} & {1,} & {0,} & {1,} & {0,} & {0,} & {0,} & {0,} & {0,} & {0,} & {0,} & {0,} & {0,} & {0,} & {1,} & {1,} & {0,} & {1,} & {1,} & {1,} & {1,} & {1,} & {0,} & {0,} \\\; & {0,} & {1,} & {0,} & {1,} & {1,} & {1,} & {0,} & {1,} & {0,} & {1,} & {1,} & {1,} & {1,} & {1,} & {1,} & {0,} & {0,} & {1,} & {1,} & {0,} & {0,} & {0,} & {0,} & {0,} \\\; & {0,} & {1,} & {0,} & {1,} & {0,} & {0,} & {0,} & {1,} & {0,} & {0,} & {0,} & {1,} & {0,} & {1,} & {0,} & {0,} & {0,} & {1,} & {0,} & {0,} & {1,} & {0,} & {1,} & {1,} \\\; & {0,} & {1,} & {1,} & {0,} & {0,} & {1,} & {1,} & {0,} & {0,} & {0,} & {1,} & {1,} & {1,} & {1,} & {1,} & {1,} & {0,} & {0,} & {0,} & {1,} & {0,} & {0,} & {0,} & {0,} \\\; & {0,} & {1,} & {1,} & \left. 1 \right\} & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \;\end{matrix}}{B^{(2)} = \begin{matrix}\left\{ {1,} \right. & {0,} & {0,} & {1,} & {1,} & {1,} & {0,} & {0,} & {1,} & {1,} & {0,} & {1,} & {1,} & {0,} & {1,} & {1,} & {1,} & {1,} & {0,} & {0,} & {0,} & {0,} & {1,} & {1,} & {0,} \\\; & {0,} & {0,} & {1,} & {1,} & {1,} & {0,} & {0,} & {1,} & {0,} & {0,} & {1,} & {1,} & {1,} & {0,} & {0,} & {0,} & {1,} & {1,} & {1,} & {1,} & {0,} & {0,} & {0,} & {1,} \\\; & {0,} & {1,} & {1,} & {1,} & {0,} & {0,} & {1,} & {1,} & {0,} & {1,} & {0,} & {0,} & {0,} & {1,} & {0,} & {0,} & {1,} & {0,} & {1,} & {0,} & {0,} & {1,} & {0,} & {1,} \\\; & {0,} & {0,} & {0,} & {0,} & {1,} & {0,} & {1,} & {1,} & {1,} & {0,} & {0,} & {1,} & {0,} & {1,} & {0,} & {1,} & {0,} & {0,} & {0,} & {1,} & {1,} & {0,} & {0,} & {1,} \\\; & {1,} & {0,} & {0,} & \left. 1 \right\} & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \;\end{matrix}}{B^{(3)} = \begin{matrix}\left\{ {1,} \right. & {0,} & {0,} & {0,} & {0,} & {1,} & {0,} & {0,} & {1,} & {1,} & {0,} & {0,} & {1,} & {0,} & {1,} & {1,} & {0,} & {1,} & {0,} & {1,} & {0,} & {0,} & {0,} & {1,} & {0,} \\\; & {1,} & {0,} & {0,} & {0,} & {0,} & {0,} & {1,} & {0,} & {1,} & {0,} & {1,} & {0,} & {0,} & {0,} & {1,} & {0,} & {1,} & {1,} & {1,} & {0,} & {0,} & {0,} & {1,} & {0,} \\\; & {0,} & {1,} & {1,} & {0,} & {1,} & {0,} & {0,} & {0,} & {0,} & {0,} & {0,} & {0,} & {1,} & {1,} & {0,} & {0,} & {1,} & {0,} & {1,} & {1,} & {1,} & {0,} & {1,} & {1,} \\\; & {0,} & {1,} & {1,} & {0,} & {1,} & {1,} & {1,} & {1,} & {1,} & {0,} & {1,} & {1,} & {1,} & {0,} & {0,} & {0,} & {0,} & {0,} & {0,} & {0,} & {1,} & {1,} & {0,} & {0,} \\\; & {0,} & {1,} & {0,} & \left. 0 \right\} & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \;\end{matrix}}}} & (1)\end{matrix}$

With r=3, the number of exclusive-or cardinality estimation combinationspossible is 2³−1=7. Table 4 shows all seven combinations for bit-wisemodulo 2 addition, along with summary statistics indicating thecorresponding count of 1s (c₁) and the estimate of {circumflex over (X)}of the corresponding exclusive-or cardinality (determined by evaluatingEquation 9) alongside the associated true (but unknown) value X.

TABLE 4 Estimated and True Values for All Combinations of Exclusive-OrCardinalities Combination c₁ (count of 1s) {circumflex over (X)}(estimated) X (truth) B⁽¹⁾ 46 40.2969 48 B⁽²⁾ 49 58.6065 41 B⁽³⁾ 4331.7834 37 B⁽¹⁾ ⊕ B⁽²⁾ 47 44.4854 35 B⁽¹⁾ ⊕ B⁽³⁾ 49 58.6065 45 B⁽²⁾ ⊕B⁽³⁾ 42 29.6975 50 B⁽¹⁾ ⊕ B⁽²⁾ ⊕ B⁽³⁾ 40 26.1758 44 Sum 289.653 300Cardinality {circumflex over (N)} = 72.413 N = 75

As shown in Table 4, the total cardinality estimate {circumflex over(N)} corresponds to the sum of the exclusive-or cardinality estimates{circumflex over (X)} divided by 4 as defined in Equation 20 for r=3.Estimates of intersections, disjoint cardinalities, or other quantitiescan also be determined. However, as the above example is based onrelatively small cardinalities and relatively short Bloom filter arrays,the errors of such estimates may be relatively large.

As the true disjoint cardinalities are known, by construction in theabove example, a simulation of multiple experiments of the above can bemade to determine some statistical properties of the estimate. A MonteCarlo experiment of 10,000 simulations yielded an estimate of theexpected value and standard deviation of {circumflex over (N)} being75.8209 and 10.952, respectively. As can be seen, the estimate of thesample expected value is close to the true cardinality of 75.

As indicated in Equation 14, there are 2^(r)−1 equations defining eachexclusive-or cardinality (e.g., X_({1}), X_({2}), X_({1,2})).Furthermore, as shown in Equation 14, these 2^(r)−1 equations aredefined based on 2^(r)−1 disjoint cardinalities (e.g., n₀₁, n₁₀, n₁₁).Thus, if each exclusive-or cardinality can be estimated (e.g., byevaluating Equation 9), it is possible to establish a full rank linearsystem to solve for each of the disjoint cardinalities. With thedisjoint cardinalities, any Boolean expression corresponding to usermembership across one or more of the r datasets is possible. Forpurposes of illustration, the liner systems relating the disjointcardinalities to the exclusive-or cardinalities are shown below in fullfor r={1, 2, 3}, with r=1 being trivially true but shown forcompleteness.

$\begin{matrix}{{\lbrack 1\rbrack\left\lbrack n_{1} \right\rbrack} = {\left\lbrack X_{\{ 1\}} \right\rbrack\mspace{31mu}\left( {r = 1} \right)}} & {{Eq}.\mspace{14mu} 26} \\{{\begin{bmatrix}0 & 1 & 1 \\1 & 0 & 1 \\1 & 1 & 0\end{bmatrix}\begin{bmatrix}n_{01} \\n_{10} \\n_{11}\end{bmatrix}} = {\begin{bmatrix}X_{\{ 1\}} \\X_{\{ 2\}} \\X_{\{{1,2}\}}\end{bmatrix}\mspace{31mu}\left( {r = 2} \right)}} & {{Eq}.\mspace{14mu} 25} \\{{\left\lbrack \begin{matrix}0 & 0 & 0 & 1 & 1 & 1 & 1 \\0 & 1 & 1 & 0 & 0 & 1 & 1 \\1 & 0 & 1 & 0 & 1 & 0 & 1 \\0 & 1 & 1 & 1 & 1 & 0 & 0 \\1 & 0 & 1 & 1 & 0 & 1 & 0 \\1 & 1 & 0 & 0 & 1 & 1 & 0 \\1 & 1 & 0 & 1 & 0 & 0 & 1\end{matrix} \right\rbrack\begin{bmatrix}n_{001} \\n_{010} \\n_{011} \\n_{100} \\n_{101} \\n_{110} \\n_{111}\end{bmatrix}} = {\begin{bmatrix}X_{\{ 1\}} \\X_{\{ 2\}} \\X_{\{ 3\}} \\X_{\{{1,2}\}} \\X_{\{{1,3}\}} \\X_{\{{2,3}\}} \\X_{\{{1,2.3}\}}\end{bmatrix}\mspace{14mu}\left( {r = 3} \right)}} & {{Eq}.\mspace{14mu} 26}\end{matrix}$

The above linear systems have applications in which specific databaseproprietors 106 a-b provide the true cardinality associated with themodulo 2 Bloom filter arrays also provided by the database proprietors.That is, as mentioned above in connection with Equation 22, the truecardinalities for each Bloom filter array may not be known butcorrespond to the summation of all disjoint cardinalities associatedwith each Bloom filter array. Thus, if the true cardinality is known(e.g., provided by the database proprietors 106 a-b), the estimates ofthe cardinalities (in the first three rows of the third column in Table4 determined based on Equation 9) may be replaced by the truecardinalities (in the first three rows of the fourth column in Table 4).In the above example summarized in Table 4, if the true values for thecardinalities of the three separate Bloom filter arrays is used in thismanner, the final estimate of the total cardinality across all threeBloom filter arrays would become {circumflex over (N)}=71.2413.

The examples described above assume that each database proprietor 106a-b provides a single modulo 2 Bloom filter array that the AME 102 maythen analyze in combination. However, in some examples, each databaseproprietor 106 a-b may generate a group of multiple modulo 2 Bloomfilter arrays. A group of multiple Bloom filter arrays may be generatedto reduce the size of each individual Bloom filter array. That is, twosmaller Bloom filter arrays may contain the same amount of informationas one larger Bloom filter array. In some examples, the different Bloomfilter arrays in the group have the same length m, but differ from oneanother in that different hash functions are used to assign users to theelements in the respective Bloom filter arrays. In such examples,Equation 9 may still be used with p=1/m. However, the value of c₁ is nolonger the count of is in a single array. Rather, the value of c₁ usedin such examples is the average of the counts across the multiple Bloomfilter arrays in the group from the corresponding database proprietor106 a-d.

As can be seen with reference to Equation 8, the largest possiblecardinality estimate for a given Bloom filter array occurs when|c₀−c₁|=1, which corresponds to when the number of 1s and 0s in thearray only differ by one. This situation arises when the Bloom filterarray is maximally mixed between elements being assigned an even or oddnumber of times. This extreme case reduces Equation 8 to

$\begin{matrix}{{\hat{n}}_{\max} = {\left( \frac{1}{k} \right)\frac{\ln\left( \frac{1}{m} \right)}{\ln\left( {1 - {2}} \right)}}} & {{Eq}.\mspace{14mu} 27}\end{matrix}$With p=1/m, the minimum array length m can be solved for that gives themaximum possible estimate of n. For large enough m (relative to n), theright hand side of Equation 27 can be approximated as

$\begin{matrix}{{\hat{n}}_{\max}\text{∼}\frac{m{\ln(m)}}{2k}} & {{Eq}.\mspace{14mu} 28}\end{matrix}$Equation 28 can be solved for m, given n and k, yielding

$\begin{matrix}{m\text{∼}\frac{2nk}{W\left( {2nk} \right)}} & {{Eq}.\mspace{14mu} 29}\end{matrix}$where W(z) is the Lambert W function, defined as the principal solutionfor w in the equation z=we^(w). This allows an estimate of the shortestarray length that could produce an estimate equal to the value of n.Shorter bit arrays will have n_(max)<n and, if n is the truecardinality, the final estimate will be biased downwards. If n_(max)>n,the individual exclusive-or estimates would be balanced in some sense inthat the over-estimated values will be offset by the under-estimatedvalues.

By way of example, given that n=10⁶ and a modulo 2 Bloom filter array isto be constructed with users assigned to individual elements threedifferent times using k=3 hash functions, the length of the bit array,according to Equation 29, would need to be m≥460, 147.33 in order forn=10⁶ to be even possibly estimated. Notably, the length of the bitarray that satisfies Equation 27, for n=10⁶, is m≥460, 148.26, therebyindicating that the approximation of Equation 29 is relatively accurate.While the above example identifies the shortest suitable array lengthfor a given cardinality, in some examples, the array length may bedefined to be longer to reduce any overall bias or error. Assuming aBloom filter array begins with all values set to 0 and the array lengthm=(5/4)nk, a large n would produce, on average, a 60:40 split betweenvalues of the array being 0 and 1 respectively. This provides a quickfirst-order approximation for the array length m, given an initialestimate of n, so that there is roughly a 50/50 split between 0s and 1sin the array (e.g., not too under-saturated, and not too over-saturated.

The foregoing examples assume that the probability that any particularuser is assigned to any particular element in a Bloom filter array isuniform across all elements in the Bloom filter array. That is, theprobability of assignment to any particular element is the same as forany or element such that p=1/m. However, the probability distributionneed not be uniform but could be based on any suitable distribution.That is, the probability of the ith element in a Bloom filter array isassigned a particular user may be defined as p=ƒ(i) for some function ofi. In such examples, the probability of the number of assignment to anygiven element, p_(E), would also be index dependent, which may beexpressed as follow for k hash functions

_(E) ^({i})½(1+(1−2p _(i))^((nk))) i∈{1, . . . ,m}  Eq. 30From Equation 30, p_(O) ^({i}) may be derived because p_(E) ^({i}) andp_(O) ^({i}) must sum to 100%. The likelihood of an assignment to anyparticular element becomes a product across all indices,

⁢( n ❘ { B i } , { p i } , k } ) = c ⁢ ∏ i = 1 m ⁢ ( p E { i } ) [ B i = 0] ⁢ ( O { i } ) [ B i = 1 ] Eq . ⁢ 31where c is a constant independent of n and does not contribute to themaximum likelihood estimation, and [A] is the Iverson bracket that has avalue equal to 1 if the state A is true and 0 otherwise. Thelog-likelihood turns into a sum

⁢( n | { B i } , { p i } , k } ) = c ⁢ ∑ i = 1 m ⁢ [ B i = 0 ] ⁢ log ⁡ ( E {i } ) + [ B i = 1 ] ⁢ log ⁡ ( p O { i } ) Eq . ⁢ 32which must be solved numerically for n.

The estimation of the total unique cardinality across r datasets asdefined in Equation 20 is still valid for Bloom filter arrays based onnon-uniform allocation of users across the elements. However, ratherthan solving for {circumflex over (X)} directly (using Equation 9),{circumflex over (X)} is solved for by maximizing the log-likelihoodnumerically from Equation 32 for each bit-wise modulo 2 addition (eachcombination of one or more of the r datasets). As a specific example,assume that p_(i) follows a geometric distribution with parameter p₀.That is,

_(i)=(1−p ₀)^((i-1)) p ₀  Eq. 33Based on this example, if p₀=0.01 and n=100 with k=1, then p_(E) ^({i})for i={1, 10, 100} would equal {0.56631, 0.579097, 0.738031}.

As mentioned above, modulo 2 Bloom filter arrays generated in accordancewith teachings disclosed herein provide greater privacy than traditionalBloom filter arrays because neither the absence nor the presence of aparticular user within an underlying dataset can be confirmed withcertainty. In some instances, to increase the privacy of traditionalBloom filter arrays, after all users have been allocated to theirrespective elements in the array, noise may be added to the array byflipping the value of ones of the elements. Inasmuch as modulo 2 Bloomfilter arrays provide privacy without the addition of noise, the processto generate modulo 2 Bloom filter arrays is an improvement in processorefficiency relative to traditional Bloom filter array generation. Insome examples, database proprietors 106 a-b may nevertheless choose toadd noise to modulo 2 Bloom filter arrays to further increase theprotection of the privacy of the users represented in the Bloom filterarrays.

Examples disclosed herein may estimate the cardinality across multiplemodulo 2 Bloom filter arrays {B⁽¹⁾, . . . , B^((r))} in which noise hasbeen added according to a random Bernoulli process with the jth Bloomfilter array having Bernoulli parameter p_(j). The noise may be added inany suitable matter during the process of generating the noisy Bloomfilter array. For instance, for different approaches, which haveequivalent outcomes, include: (1) starting with zero-valued bit-array oflength m, the ith bit is incremented by one with probability pj to addnoise, after which the allocation of users to the array follows themodulo 2 methodology outlined above; (2) starting with zero-valuedbit-array of length m, first allocate all users following the modulo 2methodology outlined above, and then add a count of one to the ith bitwith probability p_(j) with result reported using modulo 2 addition; (3)starting with zero-valued bit-array of length m, first allocate allusers following the modulo 2 methodology outlined above, and then flipthe value of the ith element with probability to add noise; and (4)instead of a zero-valued bit-array, the initialization is a randomindependent and identically distributed (IID) sample of size m accordingto the Bernoulli (p_(j)) distribution, after which allocation of usersfollows the modulo 2 methodology outlined above.

Estimating the unique cardinality across multiple modulo 2 Bloom filterarrays with Bernoulli noise is based on solving a problem dealing with acollection of biased coins. In particular, given n different coins eachwith possibly different probabilities of heads {p₁, . . . , p_(n)},where all coins are flipped once, the probability there will be an evennumber of heads observed may be expressed as follows:

P ⁢ r ⁡ ( X = { 0 , 2 , 4 , … ⁢ , n } ) = 1 2 ⁢ ( 1 + ∏ i = 1 n ⁢ ( 1 - 2 ⁢ i) ) Eq . ⁢ 34where X is the random variable of the number of heads. As a specificexample, consider the scenario where a first coin has a bias of p₁, asecond coin has a bias of p₂, and n coins all have the same bias p. Theprobability an even number of heads will be observed among thiscollection of n+2 coins, if each coin is flipped once, is the following:

_(E)=½(1+(1−2

₁)(1−2

₂)(1−2

)^(n))  Eq. 35

The above example is similar to adding noise to a modulo 2 Bloom filterarray in that the allocation of users to a particular element in theBloom filter array is comparable to flipping n coins with the sameprobability of heads being p=1/m, and the addition of noise iscomparable to flipping one other coin with some independent probabilityof heads p_(j) Thus, p_(j) is comparable to p₁ in Equation 35 with thep₂ term being dropped out. When no noise is included, the p₁ term alsodrops out to result in

_(E)=½(1+(1−2

)^(n))  Eq. 36which is the same as Equation 1 discussed above with the maximumlikelihood solution being defined by Equation 9. Thus, the addition ofBernoulli noise introduces a multiplicative constant within Equation 36labelled as d in the following expression:

_(E)=½(1+d(1−2

)^(n))  Eq. 37By analogy to the coin example described above in connection withEquation 35, it can be seen that d=1−2p_(j).

With the inclusion of this constant to account for the addition ofnoise, the maximum likelihood solution for the cardinality n (defined inEquation 9 for the no noise scenario) becomes

$\begin{matrix}{\hat{n} = {\left( \frac{1}{k} \right)\frac{\ln\left( {{\frac{1}{d}\left( {1 - \frac{2c_{1}}{m}} \right)}} \right)}{\ln\left( {1 - {2}} \right)}}} & {{Eq}.\mspace{14mu} 38}\end{matrix}$The absolute value within the logarithm is needed due to symmetry asexplained above in connection with Equation 9. Furthermore, there isanother symmetry between p_(j) and 1−p_(j) bit flipping probabilitiesthat is also valid and the reason why d is also contained in theabsolute value.

The expression for d is different when multiple Bloom filter arrays areconsidered together using bit-wise modulo 2 arithmetic across theirarrays (as described above in connection with Equation 12). Inparticular, consider any subset of the arrays expressed as the set {i}.The bit-flipping noise added to each Bloom filter array within thesubset is equivalent to one more possible additional allocationaccording to their own respective probability. This results in

d { i } = ∏ j ∈ { i } ⁢ ( 1 - 2 ⁢ j ) Eq . ⁢ 39where the product is taken across all Bloom filter arrays within thesubset given by the set {i}. Following the same maximization oflikelihood, the estimate of the exclusive-or cardinality is given as

$\begin{matrix}{{\hat{X}}_{\{ i\}} = {\left( \frac{1}{k} \right)\frac{\ln\left( {{\frac{1}{d_{\{ i\}}}\left( {1 - \frac{2c_{1}}{m}} \right)}} \right)}{\ln\left( {1 - {2}} \right)}}} & {{Eq}.\mspace{14mu} 40}\end{matrix}$where c₁ is now the number of 1's in the bit-wise modulo 2 additionacross the arrays given in the subset {i}. The cardinality estimation{circumflex over (N)} across all Bloom filter arrays is determined inaccordance with Equation 20 outlined above.

The above examples for adding noise can be generalized further. Inparticular, in some examples, instead of Bernoulli (p_(j)) for each bitof Bloom filter array j, the probability is repeating m_(j) timescreating Bernoulli (m_(j), p_(j)). This is comparable to either addingm_(j) coins with probability p_(j), or m_(j) possible bit-flipping noiseeach with the probability p_(j). The only modification for thisgeneration is that the term inside the parenthesis within Equation 39 isexponentiated to the m_(j) power.

For purposes of explanation, consider the example described aboveinvolving three different Bloom filter arrays {B⁽¹⁾, B⁽²⁾, B⁽³⁾} oflength m=101 in which k=3 hash functions were used and the total uniqueaudience (e.g., cardinality) across all three Bloom filter arrays is 75.Further, as above, the disjoint cardinalities between the three Bloomfilter arrays is defined in Equation 21 and the total number ofindividuals represented in each Bloom filter array (e.g., the truecardinality of each Bloom filter array) is defined in Equation 22. Inthis example, further assume that noise was added to each of the threeBloom filter arrays by probabilistically bit-flipping the value of eachelement in the respective Bloom filter array with given and knownprobabilities {p₁, p₂, p₃}={0.10, 0.15. 0.20}. That is, on average, 10%of the bits in the first Bloom filter array would be flipped, 15% of thebits in the second Bloom filter array would be flipped, and 20% of thebits in the third Bloom filter array would be flipped. While theseproportions of bits being flipped is expected on average, it does notfollow that exactly 10%, 15% and 20% of the bits were actually flippedin the corresponding Bloom filter arrays. Bit flipping of elementsaccording to the above probabilities was simulated for the three exampleBloom filter arrays defined above in Equation 23. The same bit arrays aswell as the resulting arrays with noise added are shown below in Table5. Further, the bits that were flipped due to the addition of noise aredemarcated via bolding and underlining.

TABLE 5 Example Bloom Filter Arrays Before and After the Addition ofNoise Original With Noise Added i B⁽¹⁾ B⁽²⁾ B⁽³⁾ B⁽¹⁾ B⁽²⁾ B⁽³⁾ 1 1 1 11 1 0 2 0 0 0 1 0 1 3 1 0 0 1 0 1 4 0 1 0 0 1 0 5 1 1 0 1 1 1 6 0 1 1 01 0 7 0 0 0 0 0 0 8 0 0 0 0 0 0 9 0 1 1 0 1 1 10 0 1 1 0 1 1 11 0 0 0 00 0 12 0 1 0 0 1 0 13 0 1 1 0 1 1 14 0 0 0 0 0 0 15 0 1 1 0 1 1 16 1 1 11 1 1 17 1 1 0 1 1 0 18 0 1 1 0 1 1 19 1 0 0 1 0 0 20 1 0 1 1 0 1 21 1 00 1 0 1 22 1 0 0 0 0 0 23 1 1 0 1 1 0 24 0 1 1 0 1 1 25 0 0 0 0 1 0 26 00 1 0 1 1 27 1 0 0 1 0 0 28 0 1 0 0 1 1 29 1 1 0 1 1 0 30 1 1 0 1 1 0 311 0 0 1 0 0 32 0 0 1 0 0 1 33 1 1 0 1 0 0 34 0 0 1 0 0 1 35 1 0 0 1 1 036 1 1 1 1 1 1 37 1 1 0 1 1 0 38 1 1 0 1 1 0 39 1 0 0 1 0 1 40 1 0 1 1 01 41 0 0 0 0 1 0 42 0 1 1 0 1 1 43 1 1 1 1 1 1 44 1 1 1 1 1 1 45 0 1 0 11 0 46 0 0 0 1 0 1 47 0 0 0 0 0 0 48 0 0 1 0 1 0 49 0 1 0 0 0 1 50 0 0 01 0 0 51 1 1 1 1 1 1 52 0 1 1 0 1 1 53 1 1 0 1 1 0 54 0 0 1 0 0 1 55 0 00 1 0 0 56 0 1 0 0 1 0 57 1 1 0 1 1 0 58 0 0 0 0 0 1 59 0 1 0 0 1 0 60 00 0 0 0 0 61 1 0 0 1 0 0 62 0 0 1 0 0 0 63 1 1 1 1 1 1 64 0 0 0 0 0 1 650 0 0 0 0 1 66 0 1 1 0 1 1 67 1 0 0 1 0 0 68 0 1 1 0 1 0 69 0 0 1 0 1 170 1 0 1 1 0 0 71 0 1 0 0 1 0 72 1 0 1 1 0 1 73 1 1 1 1 1 0 74 0 0 0 1 11 75 1 0 1 1 1 1 76 1 0 1 1 0 1 77 0 0 0 0 0 0 78 0 1 1 0 1 1 79 1 0 1 11 0 80 1 1 1 1 1 0 81 0 1 1 0 1 1 82 0 1 1 0 1 1 83 0 0 0 0 0 0 84 1 0 11 1 1 85 1 1 1 0 1 1 86 1 0 1 1 0 1 87 1 1 0 1 1 0 88 1 0 0 0 0 0 89 1 10 1 1 0 90 0 0 0 0 0 0 91 0 0 0 0 0 0 92 0 0 0 0 1 0 93 1 1 0 1 1 1 94 01 1 0 1 0 95 0 0 1 0 0 1 96 0 0 0 0 0 0 97 0 1 0 0 1 1 98 0 1 0 0 1 0 991 0 1 1 0 0 100 1 0 0 1 0 0 101 1 1 0 1 1 0

Table 6 shows all seven combinations of the three Bloom filter arraysusing bit-wise modulo 2 addition, along with summary statisticsindicating the multiplicative constant (d_({i})), the correspondingcount of 1s (c₁), and the estimate of {circumflex over (X)} of thecorresponding exclusive-or cardinality (determined by evaluatingEquation 9) alongside the associated true (but unknown) value X.

TABLE 6 Estimated and True Values for All Combinations of Exclusive-OrCardinalities c₁ Combination d_({i}) (count of 1s) {circumflex over (X)}(estimated) X (truth) B⁽¹⁾ 0.800 49 54.8876 48 B⁽²⁾ 0.700 58 25.8391 41B⁽³⁾ 0.600 46 31.7834 37 B⁽¹⁾ ⊕ B⁽²⁾ 0.560 51 67.2528 35 B⁽¹⁾ ⊕ B⁽³⁾0.480 53 37.8606 45 B⁽²⁾ ⊕ B⁽³⁾ 0.420 48 35.6352 50 ^(B(1)) ⊕ B⁽²⁾ ⊕B⁽³⁾ 0.336 43 13.6066 44 Sum 266.865 300 Cardinality {circumflex over(N)} = 66.7163 N = 75

As shown in Table 6, the total cardinality estimate {circumflex over(N)} corresponds to the sum of all exclusive-or cardinality estimatesdivided by 4 as shown in Equation 20 for r=3, yielding {circumflex over(N)}=66.7163. As can be seen by comparison with Table 4, the addition ofnoise in the Bloom filter arrays results in a different {circumflex over(N)} than when no noise was added (e.g., {circumflex over (N)}=72.413with no noise). As the above example is based on relatively smallcardinalities and relatively Bloom filter arrays, the errors produced bythe bit-flipping noise appear relatively large. However, for largerarrays and cardinalities, the bit-flipping noise would have less impactin estimation but more impact on the look of randomness of the bits inthe arrays.

FIG. 10 is a block diagram of an example database proprietor apparatus1000. The example database proprietor apparatus 1000 of FIG. 10 maycorrespond to any one of the database proprietors 106 a-b of FIG. 1 . Asshown in the illustrated example, the database proprietor apparatus 1000includes an example user database 1002, an example communicationsinterface 1004, an example Bloom filter parameter database 1006, anexample user data analyzer 1008, an example Bloom filter array generator1010, and an example noise generator 1012.

The example user database 1002 stores user data associated with users(e.g., subscribers) registered with the database proprietor apparatus1000. In some examples, the user data includes a user identifiercorresponding to any suitable PII. The example communications interface1004 enables the database proprietor apparatus 1000 to communicate withthe AME 102.

The example Bloom filter parameter database 1006 stores the Bloom filterparameters used to define and/or generate one or more modulo 2 Bloomfilter arrays representative of the users in the user database 1002. Insome examples, some or all of the Bloom filter parameters are determinedand/or received from the AME 102 (e.g., via the communications interface1004). In some examples, some or all of the Bloom filter parameters aredetermined by one or more database proprietors 106 a-b. In someexamples, the Bloom filter parameters include one or more of a length(e.g., number of bits or element) in the Bloom filter array, theidentification of one or more hash function(s) used to map users todifferent elements of the Bloom filter array and the correspondingmapping of hash function outputs to the different elements in the Bloomfilter array (e.g., parameters defining the number of different hashfunction outputs that map to each element and the particular outputsthat map to each particular element), and/or a noise parameters defininga probability with which the value of individual elements used togenerate each Bloom filter array to ensure differential privacy for thecorresponding Bloom filter array. Regardless of how the Bloom filterparameters are set or determined (e.g., whether by the AME 102 and/orthe database proprietors 106 a-b), the Bloom filter array length, hashfunctions, and corresponding hash function output mapping are to beagreed upon by all database proprietors 106 a-b. However, each databaseproprietor 106 a-b may use a different noise parameter.

The example user data analyzer 1008 analyzes user data in the userdatabase 1002 to identify users that accessed media for which the AME102 is interested in generating audience measurement metrics. Theexample Bloom filter array generator 1010 generates modulo 2 Bloomfilter arrays based on the Bloom filter parameters and the userinformation associated with users identified by the user data analyzer1008 to be included in the filter. An example process to generate amodulo 2 Bloom filter array is detailed below in connection with FIG. 12.

The example noise generator 1012 adds noise to the Bloom filter arraysgenerated by the Bloom filter array generator 1010. Due to the modulo 2addition used when generating the Bloom filter arrays, the noisegenerator 1012 may add noise to the Bloom filter array before or afterthe Bloom filter array generator 1010 allocates users to the differentelements in the Bloom filter array.

While an example manner of implementing the database proprietorapparatus 1000 is illustrated in FIG. 10 , one or more of the elements,processes and/or devices illustrated in FIG. 10 may be combined,divided, re-arranged, omitted, eliminated and/or implemented in anyother way. Further, the example user database 1002, the examplecommunications interface 1004, the example Bloom filter parameterdatabase 1006, the example Bloom filter array generator 1010, theexample user data analyzer 1008, the example noise generator 1012and/or, more generally, the example database proprietor apparatus 1000of FIG. 3 may be implemented by hardware, software, firmware and/or anycombination of hardware, software and/or firmware. Thus, for example,any of the example user database 1002, the example communicationsinterface 1004, the example Bloom filter parameter database 1006, theexample user data analyzer 1008, the example Bloom filter arraygenerator 1010, the example noise generator 1012 and/or, more generally,the example database proprietor apparatus 1000 could be implemented byone or more analog or digital circuit(s), logic circuits, programmableprocessor(s), programmable controller(s), graphics processing unit(s)(GPU(s)), digital signal processor(s) (DSP(s)), application specificintegrated circuit(s) (ASIC(s)), programmable logic device(s) (PLD(s))and/or field programmable logic device(s) (FPLD(s)). When reading any ofthe apparatus or system claims of this patent to cover a purely softwareand/or firmware implementation, at least one of the example userdatabase 1002, the example communications interface 1004, the exampleBloom filter parameter database 1006, the example user data analyzer1008, the example Bloom filter array generator 1010, and/or the examplenoise generator 1012 is/are hereby expressly defined to include anon-transitory computer readable storage device or storage disk such asa memory, a digital versatile disk (DVD), a compact disk (CD), a Blu-raydisk, etc. including the software and/or firmware. Further still, theexample database proprietor apparatus 1000 may include one or moreelements, processes and/or devices in addition to, or instead of, thoseillustrated in FIG. 10 , and/or may include more than one of any or allof the illustrated elements, processes and devices. As used herein, thephrase “in communication,” including variations thereof, encompassesdirect communication and/or indirect communication through one or moreintermediary components, and does not require direct physical (e.g.,wired) communication and/or constant communication, but ratheradditionally includes selective communication at periodic intervals,scheduled intervals, aperiodic intervals, and/or one-time events.

FIG. 11 is a block diagram of an AME apparatus 1100 of the AME 102 ofFIG. 1 . The example AME 102 includes an example audience populationanalyzer 1102, an example communications interface 1104, an exampleBloom filter parameter analyzer 1106, an example Bloom filter parameterdatabase 1108, an example Bloom filter array analyzer 1110, and anexample report generator 1112.

The example audience population analyzer 1102 determines a universeestimate for the size of population that may potentially be reached by aparticular media based on the geographic region where the media isdistributed, the platforms through which the media is distributed,and/or any other suitable factor(s). The example communicationsinterface 1104 enables the AME apparatus 1100 to communicate with thedatabase proprietors 106 a-b.

The example Bloom filter parameter analyzer 1106 determines suitableparameters for Bloom filter arrays based on the universe estimate of theaudience population analyzer 1102. More particularly, in some examples,the length of a Bloom filter array is determined based on a maximumexpected number of users in an underlying dataset to be represented inthe Bloom filter array. In some examples, the expected number of usersis determined based on the universe estimate. Further, the example Bloomfilter parameter analyzer 1106 determines parameters defining the hashfunction(s) used to evaluate PII data associated with particular usersto be represented in the Bloom filter array. Further still, in someexamples, the Bloom filter parameter analyzer 1106 determines parametersdefining how outputs of the hash functions map to particular bits orelements of the Bloom filter array. In some examples, the parametersdefining the hash function(s) and length of the Bloom filter array arestored in the Bloom filter parameter database 1108 along with otherBloom filter parameters (e.g., noise parameters). In some examples, theBloom filter parameters stored in the database 1108 may be provided tothe database proprietors 106 a-b via the example communicationsinterface 1104. In some examples, the noise parameters (and/or otherBloom filter parameters) may be provided by the database proprietors 106a-b and received via the communications interface 1104.

The example Bloom filter array analyzer 1110 analyzes Bloom filterarrays obtained from the database proprietors 106 a-b to estimate thecardinality or total number of unique users represented in individualones of the Bloom filter arrays and/or across the union of multiple suchBloom filter arrays. Further, in some examples, the Bloom filter arrayanalyzer 1110 estimates cardinalities for any Boolean combination of anintersection between different ones of the multiple Bloom filter arrays.An example process to estimate the cardinality of users across multipleBloom filter arrays is provided below in connection with FIG. 13 .

The example report generator 1112 generates any suitable reportconveying audience measurement information and estimates. In someexamples, where the Bloom filter arrays correspond to the exposure to anadvertisement in an advertising campaign, the report generated by thereport generator 1112 includes an indication of reach of the advertisingcampaign. That is, the report includes an indication of the total numberof unique individuals that were exposed to the advertisement during arelevant period of time. In some examples, the total number of uniqueindividuals corresponds to the cardinality estimate for a unioned set ofBloom filter arrays as described above.

While an example manner of implementing the AME apparatus 1100 isillustrated in FIG. 11 , one or more of the elements, processes and/ordevices illustrated in FIG. 11 may be combined, divided, re-arranged,omitted, eliminated and/or implemented in any other way. Further, theexample audience population analyzer 1102, the example communicationsinterface 1104, the example Bloom filter parameter analyzer 1106, theexample Bloom filter parameter database 1108, the example Bloom filterarray analyzer 1110, the example report generator 1112 and/or, moregenerally, the example AME apparatus 1100 of FIG. 11 may be implementedby hardware, software, firmware and/or any combination of hardware,software and/or firmware. Thus, for example, any of the example audiencepopulation analyzer 1102, the example communications interface 1104, theexample Bloom filter parameter analyzer 1106, the example Bloom filterparameter database 1108, the example Bloom filter array analyzer 1110,the example report generator 1112 and/or, more generally, the exampleAME apparatus 1100 could be implemented by one or more analog or digitalcircuit(s), logic circuits, programmable processor(s), programmablecontroller(s), graphics processing unit(s) (GPU(s)), digital signalprocessor(s) (DSP(s)), application specific integrated circuit(s)(ASIC(s)), programmable logic device(s) (PLD(s)) and/or fieldprogrammable logic device(s) (FPLD(s)). When reading any of theapparatus or system claims of this patent to cover a purely softwareand/or firmware implementation, at least one of the example audiencepopulation analyzer 1102, the example communications interface 1104, theexample Bloom filter parameter analyzer 1106, the example Bloom filterparameter database 1108, the example Bloom filter array analyzer 1110,and/or the example report generator 1112 is/are hereby expressly definedto include a non-transitory computer readable storage device or storagedisk such as a memory, a digital versatile disk (DVD), a compact disk(CD), a Blu-ray disk, etc. including the software and/or firmware.Further still, the example AME apparatus 1100 may include one or moreelements, processes and/or devices in addition to, or instead of, thoseillustrated in FIG. 11 , and/or may include more than one of any or allof the illustrated elements, processes and devices.

Flowcharts representative of example hardware logic, machine readableinstructions, hardware implemented state machines, and/or anycombination thereof for implementing the database proprietor apparatus1000 of FIG. 10 are shown in FIGS. 12 and 13 . The machine readableinstructions may be one or more executable programs or portion(s) of anexecutable program for execution by a computer processor and/orprocessor circuitry, such as the processor 1612 shown in the exampleprocessor platform 1600 discussed below in connection with FIG. 16 . Theprogram may be embodied in software stored on a non-transitory computerreadable storage medium such as a CD-ROM, a floppy disk, a hard drive, aDVD, a Blu-ray disk, or a memory associated with the processor 1612, butthe entire program and/or parts thereof could alternatively be executedby a device other than the processor 1612 and/or embodied in firmware ordedicated hardware. Further, although the example programs are describedwith reference to the flowcharts illustrated in FIGS. 12 and 13 , manyother methods of implementing the example database proprietor apparatus1000 may alternatively be used. For example, the order of execution ofthe blocks may be changed, and/or some of the blocks described may bechanged, eliminated, or combined. Additionally or alternatively, any orall of the blocks may be implemented by one or more hardware circuits(e.g., discrete and/or integrated analog and/or digital circuitry, anFPGA, an ASIC, a comparator, an operational-amplifier (op-amp), a logiccircuit, etc.) structured to perform the corresponding operation withoutexecuting software or firmware. The processor circuitry may bedistributed in different network locations and/or local to one or moredevices (e.g., a multi-core processor in a single machine, multipleprocessors distributed across a server rack, etc.).

Flowcharts representative of example hardware logic, machine readableinstructions, hardware implemented state machines, and/or anycombination thereof for implementing the AME apparatus 1100 of FIG. 11is shown in FIGS. 14 and 15 . The machine readable instructions may beone or more executable programs or portion(s) of an executable programfor execution by a computer processor and/or processor circuitry, suchas the processor 1712 shown in the example processor platform 1700discussed below in connection with FIG. 17 . The program may be embodiedin software stored on a non-transitory computer readable storage mediumsuch as a CD-ROM, a floppy disk, a hard drive, a DVD, a Blu-ray disk, ora memory associated with the processor 1712, but the entire programand/or parts thereof could alternatively be executed by a device otherthan the processor 1712 and/or embodied in firmware or dedicatedhardware. Further, although the example programs are described withreference to the flowcharts illustrated in FIGS. 14 and 15 , many othermethods of implementing the example AME apparatus 1100 may alternativelybe used. For example, the order of execution of the blocks may bechanged, and/or some of the blocks described may be changed, eliminated,or combined. Additionally or alternatively, any or all of the blocks maybe implemented by one or more hardware circuits (e.g., discrete and/orintegrated analog and/or digital circuitry, an FPGA, an ASIC, acomparator, an operational-amplifier (op-amp), a logic circuit, etc.)structured to perform the corresponding operation without executingsoftware or firmware. The processor circuitry may be distributed indifferent network locations and/or local to one or more devices (e.g., amulti-core processor in a single machine, multiple processorsdistributed across a server rack, etc.).

The machine readable instructions described herein may be stored in oneor more of a compressed format, an encrypted format, a fragmentedformat, a compiled format, an executable format, a packaged format, etc.Machine readable instructions as described herein may be stored as dataor a data structure (e.g., portions of instructions, code,representations of code, etc.) that may be utilized to create,manufacture, and/or produce machine executable instructions. Forexample, the machine readable instructions may be fragmented and storedon one or more storage devices and/or computing devices (e.g., servers)located at the same or different locations of a network or collection ofnetworks (e.g., in the cloud, in edge devices, etc.). The machinereadable instructions may require one or more of installation,modification, adaptation, updating, combining, supplementing,configuring, decryption, decompression, unpacking, distribution,reassignment, compilation, etc. in order to make them directly readable,interpretable, and/or executable by a computing device and/or othermachine. For example, the machine readable instructions may be stored inmultiple parts, which are individually compressed, encrypted, and storedon separate computing devices, wherein the parts when decrypted,decompressed, and combined form a set of executable instructions thatimplement one or more functions that may together form a program such asthat described herein.

In another example, the machine readable instructions may be stored in astate in which they may be read by processor circuitry, but requireaddition of a library (e.g., a dynamic link library (DLL)), a softwaredevelopment kit (SDK), an application programming interface (API), etc.in order to execute the instructions on a particular computing device orother device. In another example, the machine readable instructions mayneed to be configured (e.g., settings stored, data input, networkaddresses recorded, etc.) before the machine readable instructionsand/or the corresponding program(s) can be executed in whole or in part.Thus, machine readable media, as used herein, may include machinereadable instructions and/or program(s) regardless of the particularformat or state of the machine readable instructions and/or program(s)when stored or otherwise at rest or in transit.

The machine readable instructions described herein can be represented byany past, present, or future instruction language, scripting language,programming language, etc. For example, the machine readableinstructions may be represented using any of the following languages: C,C++, Java, C#, Perl, Python, JavaScript, HyperText Markup Language(HTML), Structured Query Language (SQL), Swift, etc.

As mentioned above, the example processes of FIGS. 12-15 may beimplemented using executable instructions (e.g., computer and/or machinereadable instructions) stored on a non-transitory computer and/ormachine readable medium such as a hard disk drive, a flash memory, aread-only memory, a compact disk, a digital versatile disk, a cache, arandom-access memory and/or any other storage device or storage disk inwhich information is stored for any duration (e.g., for extended timeperiods, permanently, for brief instances, for temporarily buffering,and/or for caching of the information). As used herein, the termnon-transitory computer readable medium is expressly defined to includeany type of computer readable storage device and/or storage disk and toexclude propagating signals and to exclude transmission media.

“Including” and “comprising” (and all forms and tenses thereof) are usedherein to be open ended terms. Thus, whenever a claim employs any formof “include” or “comprise” (e.g., comprises, includes, comprising,including, having, etc.) as a preamble or within a claim recitation ofany kind, it is to be understood that additional elements, terms, etc.may be present without falling outside the scope of the correspondingclaim or recitation. As used herein, when the phrase “at least” is usedas the transition term in, for example, a preamble of a claim, it isopen-ended in the same manner as the term “comprising” and “including”are open ended. The term “and/or” when used, for example, in a form suchas A, B, and/or C refers to any combination or subset of A, B, C such as(1) A alone, (2) B alone, (3) C alone, (4) A with B, (5) A with C, (6) Bwith C, and (7) A with B and with C. As used herein in the context ofdescribing structures, components, items, objects and/or things, thephrase “at least one of A and B” is intended to refer to implementationsincluding any of (1) at least one A, (2) at least one B, and (3) atleast one A and at least one B. Similarly, as used herein in the contextof describing structures, components, items, objects and/or things, thephrase “at least one of A or B” is intended to refer to implementationsincluding any of (1) at least one A, (2) at least one B, and (3) atleast one A and at least one B. As used herein in the context ofdescribing the performance or execution of processes, instructions,actions, activities and/or steps, the phrase “at least one of A and B”is intended to refer to implementations including any of (1) at leastone A, (2) at least one B, and (3) at least one A and at least one B.Similarly, as used herein in the context of describing the performanceor execution of processes, instructions, actions, activities and/orsteps, the phrase “at least one of A or B” is intended to refer toimplementations including any of (1) at least one A, (2) at least one B,and (3) at least one A and at least one B.

As used herein, singular references (e.g., “a”, “an”, “first”, “second”,etc.) do not exclude a plurality. The term “a” or “an” entity, as usedherein, refers to one or more of that entity. The terms “a” (or “an”),“one or more”, and “at least one” can be used interchangeably herein.Furthermore, although individually listed, a plurality of means,elements or method actions may be implemented by, e.g., a single unit orprocessor. Additionally, although individual features may be included indifferent examples or claims, these may possibly be combined, and theinclusion in different examples or claims does not imply that acombination of features is not feasible and/or advantageous.

In some examples, the program of FIG. 12 is independently implemented byeach database proprietor 106 a-b that is to provide a Bloom filter arrayto the AME 102 in connection with a particular item of media for whichexposure metrics are desired. The program of FIG. 12 begins at block1202 where the example Bloom filter parameter database 1006 stores Bloomfilter parameter(s) to generate a modulo 2 Bloom filter array. At block1204, the example Bloom filter array generator 1010 generates a Bloomfilter array initialized to 0. In some examples, the length of the Bloomfilter array is defined by and/or agreed upon between the AME 102 andeach database proprietor 106 a-b that is to generate a Bloom filterarray based on the example process of FIG. 12 . At block 1206, theexample user data analyzer 1008 accesses user data in the user database1002. At block 1208, the example Bloom filter array generator 1010hashes a personal identifier in the user database 1002 using a hashfunction defined in the Bloom filter parameters. At block 1210, theexample Bloom filter array generator 1010 maps an output of the hash toa corresponding element in the Bloom filter array. At block 1212, theexample Bloom filter array generator 1010 flips the value of thecorresponding element. That is, if the value was previously 1, the Bloomfilter array generator 1010 sets the value to 0, and if the value waspreviously 0, the Bloom filter array generator 1010 sets the value to 1.

At block 1214, the example Bloom filter array generator 1010 determineswhether there is another user. If so, control returns to block 1208. Ifnot, control advances to block 1216 where the example Bloom filter arraygenerator 1010 determines whether there is another hash function. If so,control returns to block 1206. Otherwise, control advances to block 1218where the example noise generator 1012 adds noise to the Bloom filterarray based on a noise parameter. In some examples, noise may beunnecessary due to the nature of the modulo 2 flipping of element valuesat block 1212 such that block 1218 is omitted. At block 1220, theexample Bloom filter array generator 1010 determines whether to generateanother Bloom filter array. In some examples, multiple Bloom filterarrays may be generated for the same underlying dataset to form a groupof Bloom filter arrays of shorter length rather than a single Bloomfilter array of longer length. If another Bloom filter array is to begenerated for the same data, control returns to block 1204 to repeat theprocess. However, during each subsequent iteration of the process (e.g.,to generate a different Bloom filter array), different hash functionsare used at block 1208 so that the allocation of users to elements ineach Bloom filter array in the group will be different. If, at block1220, the example Bloom filter array generator 1010 determines not togenerate another Bloom filter array, control advances to block 1222.

At block 1222, the example communications interface 1004 transmits theBloom filter array(s) and the associated noise parameter to the AME 102.Of course, if no noise was added, the noise parameter may be omitted.However, in some examples, a noise parameter with a value of 0 may beprovided to indicate that no noise was added to the Bloom filter array.In some examples, the communications interface 1004 may also transmit aBloom filter array cardinality for each Bloom filter array in the group(or single Bloom filter array) transmitted to the AME to indicate thetotal number of users represented in the associated Bloom filterarray(s). At block 1224, the example Bloom filter array generator 1010determines whether to update the data. In some examples, data is updatedon a relatively frequent basis (e.g., once a week, once a day, etc.). Ifthe data is to be updated, control returns to block 1204 to repeat theprocess. Otherwise, the example process of FIG. 12 ends.

In some examples, the program of FIG. 13 is independently implemented byeach database proprietor 106 a-b that is to provide a Bloom filter arrayto the AME 102 in connection with a particular item of media for whichexposure metrics are desired. The program of FIG. 13 begins at block1302 where the example Bloom filter array generator 1010 generates anarray of elements, where each element in the array has a value of 0. Atblock 1304, the example user data analyzer 1008 identifies a subset ofentries in a database (e.g., the user database 1002) to be representedin the Bloom filter array. At block 1306, the example Bloom filter arraygenerator 1010 allocates ones of the entries to respective ones of theelements in the array based on a hash function. At block 1308, theexample Bloom filter array generator 1010 flips the of one of theelements between 0 and 1 in response to each successive allocation ofone of the entries to the corresponding one of the elements. That is, ifthe value was previously 1, the Bloom filter array generator 1010 setsthe value to 0, and if the value was previously 0, the Bloom filterarray generator 1010 sets the value to 1. Thereafter, the exampleprocess of FIG. 13 ends.

The program of FIG. 14 begins at block 1402 where the example audiencepopulation analyzer 1102 determines a universe estimate for an audiencesize. At block 1404, the example Bloom filter parameter analyzer 1106determines Bloom filter parameters to generate a Bloom filter array. Atblock 1406, the example communications interface 1104 transmits theBloom filter parameters to the database proprietors 106 a-b. At block1408, the example communications interface 1104 receives modulo 2 Bloomfilter arrays from the database proprietors 106 a-b. In some examples,along with the Bloom filter arrays, the database proprietors 106 a-b mayprovide a noise parameter defining the probability at which noise wasadded to the respective Bloom filter arrays. Further, in some examples,the database proprietors 106 a-b may also provide a Bloom filter arraycardinality indicating the total number of unique users represented inthe respective Bloom filter arrays.

At block 1410, the example Bloom filter array analyzer 1110 determineswhether Bloom filter array cardinalities were provided. If so, controladvances to block 1418. If not, then the example Bloom filter arrayanalyzer 1110 needs to determine the Bloom filter array cardinalities.Accordingly, control advances to block 1412 where the example Bloomfilter array analyzer 1110 determines a count (or average count) of isin the Bloom filter array(s) from each database. If each databaseproprietor 106 a-b provided only one Bloom filter array, then a simplecount of the 1s in that Bloom filter array is sufficient. However, ifthe database proprietors 106 a-b provided a group of multiple Bloomfilter arrays, then the example Bloom filter array analyzer 1110determines the count of 1s in each Bloom filter array and thendetermines the average of the count for the corresponding group of Bloomfilter arrays.

At block 1414, the example Bloom filter array analyzer 1110 determines amultiplicative constant (d_({i})) due to noise for the Bloom filterarray. In some examples, the multiplicative constant is determined byevaluating Equation 39, where p_(j) corresponds to the noise parameterprovided by the database proprietor 106 a-b. In examples where there isno noise, the noise parameter equals 0 such that the multiplicativeconstant equals 1. In some examples, where there is no noise, block 1414may be omitted. At block 1416, the example Bloom filter array analyzer1110 estimates a Bloom filter array cardinality for each of the Bloomfilter arrays. In some examples, the Bloom filter array cardinality isestimated by evaluating Equation 40 and using the count (or averagecount) of is (determined at block 1412) as the value for c₁. In someexamples, where there is no noise, the Bloom filter array cardinalitymay be estimated by evaluating Equation 9, which is similar to Equation40 except that there is no multiplicative constant to account for thenoise in Equation 9.

At block 1418, the example Bloom filter array analyzer 1110 generatesone or more array(s) corresponding to a bit-wise union of anexclusive-or combination of at least two of the Bloom filter arrays.Multiple arrays are generated for the same exclusive-or combination whena group of multiple Bloom filter arrays are provided by the databaseproprietors 106 a-b. In some examples, the bit wise union is implementedbased on modulo 2 addition as shown and described in connection withEquation 12. After the array(s) for a particular combination of theBloom filter arrays has been generated, the example Bloom filter arrayanalyzer 1110 determines the exclusive-or cardinality for the array(s)following a similar process to determine the Bloom filter arraycardinalities described above at blocks 1412-1316. That is, at block1420, the example Bloom filter array analyzer 1110 determines a count(or average count) of is in the array(s). At block 1422, the exampleBloom filter array analyzer 1110 determines a multiplicative constantdue to noise. In some examples, block 1422 may be omitted because themultiplicative constant was already determined at block 1414. At block1424, the example Bloom filter array analyzer 1110 estimates theexclusive-or cardinality for the array(s).

At block 1426, the example Bloom filter array analyzer 1110 determineswhether there is another combination of Bloom filter arrays to analyze.As described above, every combination of the Bloom filter arraysincluding taking them 1 at time, 2 at a time, up to taking all of themtogether is analyzed. The analysis of each of the Bloom filter arraysindividual (e.g., taken 1 at a time) is accomplished at blocks 1410-1316such that the determination at block 1426 relates to combinations of twoor more Bloom filter arrays. If the example Bloom filter array analyzer1110 determines that there is another combination of Bloom filter arraysto analyze, control returns to block 1418. Otherwise, control advancesto block 1428.

At block 1428, the example Bloom filter array analyzer 1110 estimatesthe overall cardinality across all the Bloom filter arrays. In someexamples, the overall cardinality is estimated by evaluating Equation20, which involves summing each of the Bloom filter array cardinalitiesand each of the exclusive-or cardinalities, and then dividing the totalby a constant. At block 1430, the example report generator 1112generates a report based on the overall cardinality estimate. At block1432, the example communications interface 1104 transmits the report toan interested third party. At block 1434, the example report generator1112 determines whether to generate an updated and/or new report. Asmentioned above, in some examples, reports are generated on a relativelyfrequent basis (e.g., weekly, daily, etc.). If an updated and/or newreport is to be generated, control returns to block 1408. Otherwise, theexample program of FIG. 14 ends.

The example program of FIG. 15 begins at block 1502 where the examplecommunications interface 1104 accesses and/or receives a first Bloomfilter array generated by a first computer of a first databaseproprietor, where the first Bloom filter array is representative offirst users who accessed media, the first users is registered with thefirst database proprietor, the first Bloom filter array includes a firstarray of first elements, and values of respective ones of the firstelements are either a 0 or a 1 based on whether quantities of the firstusers allocated to the respective ones of the first elements are even orodd. At block 1504, the example Bloom filter analyzer 1110 estimates afirst cardinality for the first Bloom filter array, where the firstcardinality indicative of a total number of the first users who accessedthe media. Thereafter, the example process of FIG. 15 ends.

FIG. 16 is a block diagram of an example processor platform 1600structured to execute the instructions of FIG. 12 to implement thedatabase proprietor apparatus 1000 of FIG. 10 . The processor platform1600 can be, for example, a server, a personal computer, a workstation,a self-learning machine (e.g., a neural network), a mobile device (e.g.,a cell phone, a smart phone, a tablet such as an iPad™), a personaldigital assistant (PDA), an Internet appliance, or any other type ofcomputing device.

The processor platform 1600 of the illustrated example includes aprocessor 1612. The processor 1612 of the illustrated example ishardware. For example, the processor 1612 can be implemented by one ormore integrated circuits, logic circuits, microprocessors, GPUs, DSPs,or controllers from any desired family or manufacturer. The hardwareprocessor may be a semiconductor based (e.g., silicon based) device. Inthis example, the processor implements the example user data analyzer1008, the example Bloom filter array generator 1010 and the examplenoise generator 1012.

The processor 1612 of the illustrated example includes a local memory1613 (e.g., a cache). The processor 1612 of the illustrated example isin communication with a main memory including a volatile memory 1614 anda non-volatile memory 1616 via a bus 1618. The volatile memory 1614 maybe implemented by Synchronous Dynamic Random Access Memory (SDRAM),Dynamic Random Access Memory (DRAM), RAMBUS® Dynamic Random AccessMemory (RDRAM®) and/or any other type of random access memory device.The non-volatile memory 1616 may be implemented by flash memory and/orany other desired type of memory device. Access to the main memory 1614,1616 is controlled by a memory controller.

The processor platform 1600 of the illustrated example also includes aninterface circuit 1620. In this example, the interface circuit 1620implements the example communications interface 1004. The interfacecircuit 1620 may be implemented by any type of interface standard, suchas an Ethernet interface, a universal serial bus (USB), a Bluetooth®interface, a near field communication (NFC) interface, and/or a PCIexpress interface.

In the illustrated example, one or more input devices 1622 are connectedto the interface circuit 1620. The input device(s) 1622 permit(s) a userto enter data and/or commands into the processor 1612. The inputdevice(s) can be implemented by, for example, an audio sensor, amicrophone, a camera (still or video), a keyboard, a button, a mouse, atouchscreen, a track-pad, a trackball, isopoint and/or a voicerecognition system.

One or more output devices 1624 are also connected to the interfacecircuit 1620 of the illustrated example. The output devices 1624 can beimplemented, for example, by display devices (e.g., a light emittingdiode (LED), an organic light emitting diode (OLED), a liquid crystaldisplay (LCD), a cathode ray tube display (CRT), an in-place switching(IPS) display, a touchscreen, etc.), a tactile output device, a printerand/or speaker. The interface circuit 1620 of the illustrated example,thus, typically includes a graphics driver card, a graphics driver chipand/or a graphics driver processor.

The interface circuit 1620 of the illustrated example also includes acommunication device such as a transmitter, a receiver, a transceiver, amodem, a residential gateway, a wireless access point, and/or a networkinterface to facilitate exchange of data with external machines (e.g.,computing devices of any kind) via a network 1626. The communication canbe via, for example, an Ethernet connection, a digital subscriber line(DSL) connection, a telephone line connection, a coaxial cable system, asatellite system, a line-of-site wireless system, a cellular telephonesystem, etc.

The processor platform 1600 of the illustrated example also includes oneor more mass storage devices 1628 for storing software and/or data.Examples of such mass storage devices 1628 include floppy disk drives,hard drive disks, compact disk drives, Blu-ray disk drives, redundantarray of independent disks (RAID) systems, and digital versatile disk(DVD) drives. In this example, the mass storage devices 1628 implementthe example user database 1002 and the example Bloom filter parameterdatabase 1006.

The machine executable instructions 1632 of FIG. 12 may be stored in themass storage device 1628, in the volatile memory 1614, in thenon-volatile memory 1616, and/or on a removable non-transitory computerreadable storage medium such as a CD or DVD.

FIG. 17 is a block diagram of an example processor platform 1700structured to execute the instructions of FIG. 14 to implement the AMEapparatus 1100 of FIG. 11 . The processor platform 1700 can be, forexample, a server, a personal computer, a workstation, a self-learningmachine (e.g., a neural network), a mobile device (e.g., a cell phone, asmart phone, a tablet such as an iPad™), a personal digital assistant(PDA), an Internet appliance, or any other type of computing device.

The processor platform 1700 of the illustrated example includes aprocessor 1712. The processor 1712 of the illustrated example ishardware. For example, the processor 1712 can be implemented by one ormore integrated circuits, logic circuits, microprocessors, GPUs, DSPs,or controllers from any desired family or manufacturer. The hardwareprocessor may be a semiconductor based (e.g., silicon based) device. Inthis example, the processor implements the example audience populationanalyzer 1102, the example Bloom filter parameter analyzer 1106, theexample Bloom filter array analyzer 1110, and the example reportgenerator 1112.

The processor 1712 of the illustrated example includes a local memory1713 (e.g., a cache). The processor 1712 of the illustrated example isin communication with a main memory including a volatile memory 1714 anda non-volatile memory 1716 via a bus 1718. The volatile memory 1714 maybe implemented by Synchronous Dynamic Random Access Memory (SDRAM),Dynamic Random Access Memory (DRAM), RAMBUS® Dynamic Random AccessMemory (RDRAM®) and/or any other type of random access memory device.The non-volatile memory 1716 may be implemented by flash memory and/orany other desired type of memory device. Access to the main memory 1714,1716 is controlled by a memory controller.

The processor platform 1700 of the illustrated example also includes aninterface circuit 1720. The interface circuit 1720 may be implemented byany type of interface standard, such as an Ethernet interface, auniversal serial bus (USB), a Bluetooth® interface, a near fieldcommunication (NFC) interface, and/or a PCI express interface.

In the illustrated example, one or more input devices 1722 are connectedto the interface circuit 1720. In this example, the interface circuit1720 implements the example communications interface 1104. The inputdevice(s) 1722 permit(s) a user to enter data and/or commands into theprocessor 1712. The input device(s) can be implemented by, for example,an audio sensor, a microphone, a camera (still or video), a keyboard, abutton, a mouse, a touchscreen, a track-pad, a trackball, isopointand/or a voice recognition system.

One or more output devices 1724 are also connected to the interfacecircuit 1720 of the illustrated example. The output devices 1724 can beimplemented, for example, by display devices (e.g., a light emittingdiode (LED), an organic light emitting diode (OLED), a liquid crystaldisplay (LCD), a cathode ray tube display (CRT), an in-place switching(IPS) display, a touchscreen, etc.), a tactile output device, a printerand/or speaker. The interface circuit 1720 of the illustrated example,thus, typically includes a graphics driver card, a graphics driver chipand/or a graphics driver processor.

The interface circuit 1720 of the illustrated example also includes acommunication device such as a transmitter, a receiver, a transceiver, amodem, a residential gateway, a wireless access point, and/or a networkinterface to facilitate exchange of data with external machines (e.g.,computing devices of any kind) via a network 1726. The communication canbe via, for example, an Ethernet connection, a digital subscriber line(DSL) connection, a telephone line connection, a coaxial cable system, asatellite system, a line-of-site wireless system, a cellular telephonesystem, etc.

The processor platform 1700 of the illustrated example also includes oneor more mass storage devices 1728 for storing software and/or data.Examples of such mass storage devices 1728 include floppy disk drives,hard drive disks, compact disk drives, Blu-ray disk drives, redundantarray of independent disks (RAID) systems, and digital versatile disk(DVD) drives. In this example, the mass storage devices 1728 implementsthe example Bloom filter parameter database 1108.

The machine executable instructions 1732 of FIG. 14 may be stored in themass storage device 1728, in the volatile memory 1714, in thenon-volatile memory 1716, and/or on a removable non-transitory computerreadable storage medium such as a CD or DVD.

From the foregoing, it will be appreciated that example methods,apparatus and articles of manufacture have been disclosed that enablethe generation of a modulo 2 Bloom filter array that provides increasedprivacy relative to traditional Bloom filter arrays because of therepeating flipping between 0s and 1s. Furthermore, the flipping between0s to 1s and from 1s back to 0s reduces the concern for saturation(substantially all elements becoming 1s) such that the length of modulo2 Bloom filter arrays described herein may have a shorter length thantraditional Bloom filter arrays. The shorter length of Bloom filterarray results in the need for less memory space and for more efficientprocessing. Thus, the disclosed methods, apparatus and articles ofmanufacture are accordingly directed to one or more improvement(s) inthe functioning of a computer. Furthermore, in addition to providingincreased privacy protection and reducing memory and processingrequirements, the modulo 2 Bloom filter arrays may also be used toestimate the cardinality indicative of the number of users within anunderlying dataset of a Bloom filter array as well as the overallcardinality across the union of multiple Bloom filter arrays. This isparticularly advantageous in the technical field of audience measurementof online media where some database proprietors are no longer supportingthird-party cookies such that audience measurement entities can nolonger track exposure to media directly, but must rely on reports fromthe database proprietors in the form of sketch data (such as Bloomfilter arrays as disclosed herein) that preserves the privacy of theirusers.

Example 1 includes an apparatus comprising a communications interface toreceive a first Bloom filter array from a first computer of a firstdatabase proprietor, the first Bloom filter array representative offirst users who accessed media, the first users registered with thefirst database proprietor, the first Bloom filter array including afirst array of first elements, values of respective ones of the firstelements being either a 0 or a 1 based on whether quantities of thefirst users allocated to the respective ones of the first elements areeven or odd, and a Bloom filter array analyzer to estimate a firstcardinality for the first Bloom filter array, the first cardinalityindicative of a total number of the first users who accessed the media.

Example 2 includes the apparatus of example 1, wherein the Bloom filterarray analyzer is to determine a count of the first elements with avalue of 1, and estimate the first cardinality based on the count.

Example 3 includes the apparatus of example 2, wherein the count is afirst count, the communications interface to receive a second Bloomfilter array from the first computer of the first database proprietor,the second Bloom filter array representative of the first users whoaccessed media, the second Bloom filter array including a second arrayof second elements, the first users allocated to ones of the firstelements of the first array based on a first hash function and allocatedto ones of the second elements of the second array based on a secondhash function different than the first has function, the Bloom filterarray analyzer to determine a second count of the second elements with avalue of 1, and estimate the first cardinality based on an average ofthe first and second counts.

Example 4 includes the apparatus of any one of examples 1-3, wherein theBloom filter array analyzer is to determine a multiplicative constantbased on a noise parameter, the noise parameter defining a probabilityat which ones of the values of respective ones of the first elements areflipped between 0 and 1 independent of an allocation of the first usersto the respective ones of the first elements, and estimate the firstcardinality based on the multiplicative constant.

Example 5 includes the apparatus of any one of examples 1-4, wherein thecommunications interface is to receive a second Bloom filter array froma second computer of a second database proprietor, the second Bloomfilter array representative of second users who accessed the media, thesecond users registered with the second database proprietor, the secondBloom filter array including a second array of second elements, valuesof respective ones of the second elements being either a 0 or a 1 basedon whether quantities of the second users allocated to the respectiveones of the second elements are even or odd, the Bloom filter arrayanalyzer to estimate an overall cardinality across both the first andsecond Bloom filter arrays, the overall cardinality indicative of atotal number of unique individuals corresponding to the first and secondusers who accessed the media.

Example 6 includes the apparatus of example 5, wherein the first arrayof first elements has a same length as the second array of secondelements, the length corresponding to an odd number of elements.

Example 7 includes the apparatus of any one of examples 5 or 6, whereinthe Bloom filter array analyzer is to generate a third array of thirdelements based on a bit-wise union of the first array and the secondarray, the bit-wise union based on modulo 2 addition, and estimate theoverall cardinality based on the third array.

Example 8 includes the apparatus of any one of examples 1-7, wherein thecommunications interface is to receive a plurality of Bloom filterarrays including the first Bloom filter array, ones of the Bloom filterarrays representative of different users who accessed the media, theBloom filter array analyzer to generate a plurality of arrays based onbit-wise unions between different sets of at least two of the pluralityof Bloom filter arrays, the bit-wise unions based on modulo 2 addition,ones of the plurality of arrays representative of exclusive-or groupingsof the different users included within datasets underlying respectiveones of the plurality of Bloom filter arrays, estimate a plurality ofexclusive-or cardinalities for the plurality of arrays, and estimate anoverall cardinality across the plurality of Bloom filter arrays based ona summation of the exclusive-or cardinalities.

Example 9 includes a non-transitory computer readable medium comprisinginstructions that, when executed, cause a machine to at least access afirst Bloom filter array generated by a first computer of a firstdatabase proprietor, the first Bloom filter array representative offirst users who accessed media, the first users registered with thefirst database proprietor, the first Bloom filter array including afirst array of first elements, values of respective ones of the firstelements being either a 0 or a 1 based on whether quantities of thefirst users allocated to the respective ones of the first elements areeven or odd, and estimate a first cardinality for the first Bloom filterarray, the first cardinality indicative of a total number of the firstusers who accessed the media.

Example 10 includes the non-transitory computer readable medium ofexample 9, wherein the instructions further cause the machine todetermine a count of the first elements with a value of 9, and estimatethe first cardinality based on the count.

Example 11 includes the non-transitory computer readable medium ofexample 10, wherein the count is a first count, and the instructionsfurther cause the machine to access a second Bloom filter arraygenerated by the first computer of the first database proprietor, thesecond Bloom filter array representative of the first users who accessedmedia, the second Bloom filter array including a second array of secondelements, the first users allocated to ones of the first elements of thefirst array based on a first hash function and allocated to ones of thesecond elements of the second array based on a second hash functiondifferent than the first has function, determine a second count of thesecond elements with a value of 1, and estimate the first cardinalitybased on an average of the first and second counts.

Example 12 includes the non-transitory computer readable medium of anyone of examples 9-11, wherein the instructions further cause the machineto determine a multiplicative constant based on a noise parameter, thenoise parameter defining a probability at which ones of the values ofrespective ones of the first elements are flipped between 0 and 1independent of an allocation of the first users to the respective onesof the first elements, and estimate the first cardinality based on themultiplicative constant.

Example 13 includes the non-transitory computer readable medium of anyone of examples 9-12, wherein the instructions further cause the machineto access a second Bloom filter array generated by a second computer ofa second database proprietor, the second Bloom filter arrayrepresentative of second users who accessed the media, the second usersregistered with the second database proprietor, the second Bloom filterarray including a second array of second elements, values of respectiveones of the second elements being either a 0 or a 1 based on whetherquantities of the second users allocated to the respective ones of thesecond elements are even or odd, and estimate an overall cardinalityacross both the first and second Bloom filter arrays, the overallcardinality indicative of a total number of unique individualscorresponding to the first and second users who accessed the media.

Example 14 includes the non-transitory computer readable medium ofexample 13, wherein the first array of first elements has a same lengthas the second array of second elements, the length corresponding to anodd number of elements.

Example 15 includes the non-transitory computer readable medium of anyone of examples 13 or 14, wherein the instructions further cause themachine to generate a third array of third elements based on a bit-wiseunion of the first array and the second array, the bit-wise union basedon modulo 2 addition, and estimate the overall cardinality based on thethird array.

Example 16 includes the non-transitory computer readable medium of anyone of examples 9-15, wherein the instructions further cause the machineto access a plurality of Bloom filter arrays including the first Bloomfilter array, ones of the Bloom filter arrays representative ofdifferent users who accessed the media, generate a plurality of arraysbased on bit-wise unions between different sets of at least two of theplurality of Bloom filter arrays, the bit-wise unions based on modulo 2addition, ones of the plurality of arrays representative of exclusive-orgroupings of the different users included within datasets underlyingrespective ones of the plurality of Bloom filter arrays, estimate aplurality of exclusive-or cardinalities for the plurality of arrays, andestimate an overall cardinality across the plurality of Bloom filterarrays based on a summation of the exclusive-or cardinalities.

Example 17 includes a method comprising accessing a first Bloom filterarray generated by a first computer of a first database proprietor, thefirst Bloom filter array representative of first users who accessedmedia, the first users registered with the first database proprietor,the first Bloom filter array including a first array of first elements,values of respective ones of the first elements being either a 0 or a 1based on whether quantities of the first users allocated to therespective ones of the first elements are even or odd, and estimating,by executing an instruction with a processor, a first cardinality forthe first Bloom filter array, the first cardinality indicative of atotal number of the first users who accessed the media.

Example 18 includes the method of example 17, further includingdetermining a count of the first elements with a value of 1, andestimating the first cardinality based on the count.

Example 19 includes the method of example 18, wherein the count is afirst count, and further including accessing a second Bloom filter arraygenerated by the first computer of the first database proprietor, thesecond Bloom filter array representative of the first users who accessedmedia, the second Bloom filter array including a second array of secondelements, the first users allocated to ones of the first elements of thefirst array based on a first hash function and allocated to ones of thesecond elements of the second array based on a second hash functiondifferent than the first has function, determining a second count of thesecond elements with a value of 1, and estimating the first cardinalitybased on an average of the first and second counts.

Example 20 includes the method of any one of examples 17-19, furtherincluding determining a multiplicative constant based on a noiseparameter, the noise parameter defining a probability at which ones ofthe values of respective ones of the first elements are flipped between0 and 1 independent of an allocation of the first users to therespective ones of the first elements, and estimating the firstcardinality based on the multiplicative constant.

Example 21 includes the method of any one of examples 17-20, furtherincluding accessing a second Bloom filter array generated by a secondcomputer of a second database proprietor, the second Bloom filter arrayrepresentative of second users who accessed the media, the second usersregistered with the second database proprietor, the second Bloom filterarray including a second array of second elements, values of respectiveones of the second elements being either a 0 or a 1 based on whetherquantities of the second users allocated to the respective ones of thesecond elements are even or odd, and estimating an overall cardinalityacross both the first and second Bloom filter arrays, the overallcardinality indicative of a total number of unique individualscorresponding to the first and second users who accessed the media.

Example 22 includes the method of example 21, wherein the first array offirst elements has a same length as the second array of second elements,the length corresponding to an odd number of elements.

Example 23 includes the method of any one of examples 21 or 22, furtherincluding generating a third array of third elements based on a bit-wiseunion of the first array and the second array, the bit-wise union basedon modulo 2 addition, and estimating the overall cardinality based onthe third array.

Example 24 includes the method of any one of examples 17-23, furtherincluding accessing a plurality of Bloom filter arrays including thefirst Bloom filter array, ones of the Bloom filter arrays representativeof different users who accessed the media, generating a plurality ofarrays based on bit-wise unions between different sets of at least twoof the plurality of Bloom filter arrays, the bit-wise unions based onmodulo 2 addition, ones of the plurality of arrays representative ofexclusive-or groupings of the different users included within datasetsunderlying respective ones of the plurality of Bloom filter arrays,estimating a plurality of exclusive-or cardinalities for the pluralityof arrays, and estimating an overall cardinality across the plurality ofBloom filter arrays based on a summation of the exclusive-orcardinalities.

Example 25 includes an apparatus to generate a modulo 2 Bloom filterarray, the method comprising a data analyzer to identify a subset ofentries in a database to be represented in the Bloom filter array, and aBloom filter array generator to generate an array of elements, eachelement in the array having a value of 0, allocate ones of the entriesto respective ones of the elements in the array based on a hashfunction, and flip the value of a first one of the elements between 0and 1 in response to each successive allocation of one of the entries tothe first one of the elements.

Example 26 includes a non-transitory computer readable medium comprisinginstructions that, when executed, cause a machine to at least identify asubset of entries in a database to be represented in the Bloom filterarray, generate an array of elements, each element in the array having avalue of 0, allocate ones of the entries to respective ones of theelements in the array based on a hash function, and flip the value of afirst one of the elements between 0 and 1 in response to each successiveallocation of one of the entries to the first one of the elements.

Example 27 includes a method to generate a modulo 2 Bloom filter array,the method comprising generating, by executing an instruction with aprocessor, an array of elements, each element in the array having avalue of 0, identifying, by executing an instruction with the processor,a subset of entries in a database to be represented in the Bloom filterarray, allocating, by executing an instruction with the processor, onesof the entries to respective ones of the elements in the array based ona hash function, and flipping, by executing an instruction with theprocessor, the value of a first one of the elements between 0 and 1 inresponse to each successive allocation of one of the entries to thefirst one of the elements.

Although certain example methods, apparatus and articles of manufacturehave been disclosed herein, the scope of coverage of this patent is notlimited thereto. On the contrary, this patent covers all methods,apparatus and articles of manufacture fairly falling within the scope ofthe claims of this patent.

The following claims are hereby incorporated into this DetailedDescription by this reference, with each claim standing on its own as aseparate embodiment of the present disclosure.

What is claimed is:
 1. An apparatus comprising: a communicationsinterface to: receive a first Bloom filter array from a first computerof a first database proprietor, the first Bloom filter arrayrepresentative of first users who accessed media, the first usersregistered with the first database proprietor, the first Bloom filterarray including a first array of first elements, values of respectiveones of the first elements being either a 0 or a 1 based on whetherquantities of the first users allocated to the respective ones of thefirst elements are even or odd; and receive a second Bloom filter arrayfrom the first computer of the first database proprietor, the secondBloom filter array representative of the first users who accessed media,the second Bloom filter array including a second array of secondelements, the first users allocated to ones of the first elements of thefirst array based on a first hash function and allocated to ones of thesecond elements of the second array based on a second hash functiondifferent than the first hash function; and a Bloom filter arrayanalyzer to: determine a first count of the first elements with a valueof 1; determine a second count of the second elements with a value of 1;and estimate a first cardinality for the first Bloom filter array basedon an average of the first and second counts, the first cardinalityindicative of a total number of the first users who accessed the media.2. The apparatus of claim 1, wherein the Bloom filter array analyzer isto: determine a multiplicative constant based on a noise parameter, thenoise parameter defining a probability at which ones of the values ofrespective ones of the first elements are flipped between 0 and 1independent of an allocation of the first users to the respective onesof the first elements; and estimate the first cardinality based on themultiplicative constant.
 3. An apparatus comprising: a communicationsinterface to: receive a first Bloom filter array from a first computerof a first database proprietor, the first Bloom filter arrayrepresentative of first users who accessed media, the first usersregistered with the first database proprietor, the first Bloom filterarray including a first array of first elements, values of respectiveones of the first elements being either a 0 or a 1 based on whetherquantities of the first users allocated to the respective ones of thefirst elements are even or odd; and receive a second Bloom filter arrayfrom a second computer of a second database proprietor, the second Bloomfilter array representative of second users who accessed the media, thesecond users registered with the second database proprietor, the secondBloom filter array including a second array of second elements, valuesof respective ones of the second elements being either a 0 or a 1 basedon whether quantities of the second users allocated to the respectiveones of the second elements are even or odd; and a Bloom filter arrayanalyzer to: estimate an overall cardinality across both the first andsecond Bloom filter arrays, the overall cardinality indicative of atotal number of unique individuals corresponding to the first and secondusers who accessed the media.
 4. The apparatus of claim 3, wherein thefirst array of first elements has a same length as the second array ofsecond elements, the length corresponding to an odd number of elements.5. The apparatus of claim 3, wherein the Bloom filter array analyzer isto: generate a third array of third elements based on a bit-wise unionof the first array and the second array, the bit-wise union based onmodulo 2 addition; and estimate the overall cardinality based on thethird array.
 6. An apparatus comprising: a communications interface isto receive a plurality of Bloom filter arrays including a first Bloomfilter array, the first Bloom filter array received from a firstcomputer of a first database proprietor, the first Bloom filter arrayrepresentative of first users who accessed media, the first usersregistered with the first database proprietor, the first Bloom filterarray including a first array of first elements, values of respectiveones of the first elements being either a 0 or a 1 based on whetherquantities of the first users allocated to the respective ones of thefirst elements are even or odd, ones of the plurality of Bloom filterarrays representative of different users who accessed the media; and aBloom filter array analyzer to: generate a plurality of arrays based onbit-wise unions between different sets of at least two of the pluralityof Bloom filter arrays, the bit-wise unions based on modulo 2 addition,ones of the plurality of arrays representative of exclusive-or groupingsof the different users included within datasets underlying respectiveones of the plurality of Bloom filter arrays; estimate a plurality ofexclusive-or cardinalities for the plurality of arrays; and estimate anoverall cardinality across the plurality of Bloom filter arrays based ona summation of the exclusive-or cardinalities.
 7. A non-transitorycomputer readable medium comprising instructions that, when executed,cause a machine to at least: access a first Bloom filter array generatedby a first computer of a first database proprietor, the first Bloomfilter array representative of first users who accessed media, the firstusers registered with the first database proprietor, the first Bloomfilter array including a first array of first elements, values ofrespective ones of the first elements being either a 0 or a 1 based onwhether quantities of the first users allocated to the respective onesof the first elements are even or odd; access a second Bloom filterarray generated by the first computer of the first database proprietor,the second Bloom filter array representative of the first users whoaccessed media, the second Bloom filter array including a second arrayof second elements, the first users allocated to ones of the firstelements of the first array based on a first hash function and allocatedto ones of the second elements of the second array based on a secondhash function different than the first hash function; determine a firstcount of the first elements with a value of 1; determine a second countof the second elements with a value of 1; and estimate a firstcardinality for the first Bloom filter array based on an average of thefirst and second counts, the first cardinality indicative of a totalnumber of the first users who accessed the media.
 8. The non-transitorycomputer readable medium of claim 7, wherein the instructions furthercause the machine to: determine a multiplicative constant based on anoise parameter, the noise parameter defining a probability at whichones of the values of respective ones of the first elements are flippedbetween 0 and 1 independent of an allocation of the first users to therespective ones of the first elements; and estimate the firstcardinality based on the multiplicative constant.
 9. A non-transitorycomputer readable medium comprising instructions that, when executed,cause a machine to at least: access a first Bloom filter array generatedby a first computer of a first database proprietor, the first Bloomfilter array representative of first users who accessed media, the firstusers registered with the first database proprietor, the first Bloomfilter array including a first array of first elements, values ofrespective ones of the first elements being either a 0 or a 1 based onwhether quantities of the first users allocated to the respective onesof the first elements are even or odd; access a second Bloom filterarray generated by a second computer of a second database proprietor,the second Bloom filter array representative of second users whoaccessed the media, the second users registered with the second databaseproprietor, the second Bloom filter array including a second array ofsecond elements, values of respective ones of the second elements beingeither a 0 or a 1 based on whether quantities of the second usersallocated to the respective ones of the second elements are even or odd;and estimate an overall cardinality across both the first and secondBloom filter arrays, the overall cardinality indicative of a totalnumber of unique individuals corresponding to the first and second userswho accessed the media.
 10. The non-transitory computer readable mediumof claim 9, wherein the first array of first elements has a same lengthas the second array of second elements, the length corresponding to anodd number of elements.
 11. The non-transitory computer readable mediumof claim 9, wherein the instructions further cause the machine to:generate a third array of third elements based on a bit-wise union ofthe first array and the second array, the bit-wise union based on modulo2 addition; and estimate the overall cardinality based on the thirdarray.
 12. A non-transitory computer readable medium comprisinginstructions that, when executed, cause a machine to at least: access aplurality of Bloom filter arrays including a first Bloom filter array,the first Bloom filter array generated by a first computer of a firstdatabase proprietor, the first Bloom filter array representative offirst users who accessed media, the first users registered with thefirst database proprietor, the first Bloom filter array including afirst array of first elements, values of respective ones of the firstelements being either a 0 or a 1 based on whether quantities of thefirst users allocated to the respective ones of the first elements areeven or odd, ones of the plurality of Bloom filter arrays representativeof different users who accessed the media; generate a plurality ofarrays based on bit-wise unions between different sets of at least twoof the plurality of Bloom filter arrays, the bit-wise unions based onmodulo 2 addition, ones of the plurality of arrays representative ofexclusive-or groupings of the different users included within datasetsunderlying respective ones of the plurality of Bloom filter arrays;estimate a plurality of exclusive-or cardinalities for the plurality ofarrays; and estimate an overall cardinality across the plurality ofBloom filter arrays based on a summation of the exclusive-orcardinalities.
 13. A method comprising: accessing a first Bloom filterarray generated by a first computer of a first database proprietor, thefirst Bloom filter array representative of first users who accessedmedia, the first users registered with the first database proprietor,the first Bloom filter array including a first array of first elements,values of respective ones of the first elements being either a 0 or a 1based on whether quantities of the first users allocated to therespective ones of the first elements are even or odd; accessing asecond Bloom filter array generated by the first computer of the firstdatabase proprietor, the second Bloom filter array representative of thefirst users who accessed media, the second Bloom filter array includinga second array of second elements, the first users allocated to ones ofthe first elements of the first array based on a first hash function andallocated to ones of the second elements of the second array based on asecond hash function different than the first hash function; determininga first count of the first elements with a value of 1; determining asecond count of the second elements with a value of 1; and estimating,by executing an instruction with a processor, a first cardinality forthe first Bloom filter array based on an average of the first and secondcounts, the first cardinality indicative of a total number of the firstusers who accessed the media.
 14. The method of claim 13, furtherincluding: determining a multiplicative constant based on a noiseparameter, the noise parameter defining a probability at which ones ofthe values of respective ones of the first elements are flipped between0 and 1 independent of an allocation of the first users to therespective ones of the first elements; and estimating the firstcardinality based on the multiplicative constant.
 15. A methodcomprising: accessing a first Bloom filter array generated by a firstcomputer of a first database proprietor, the first Bloom filter arrayrepresentative of first users who accessed media, the first usersregistered with the first database proprietor, the first Bloom filterarray including a first array of first elements, values of respectiveones of the first elements being either a 0 or a 1 based on whetherquantities of the first users allocated to the respective ones of thefirst elements are even or odd; accessing a second Bloom filter arraygenerated by a second computer of a second database proprietor, thesecond Bloom filter array representative of second users who accessedthe media, the second users registered with the second databaseproprietor, the second Bloom filter array including a second array ofsecond elements, values of respective ones of the second elements beingeither a 0 or a 1 based on whether quantities of the second usersallocated to the respective ones of the second elements are even or odd;estimating, by executing an instruction with a processor, a firstcardinality for the first Bloom filter array, the first cardinalityindicative of a total number of the first users who accessed the media;and estimating, by executing an instruction with the processor, anoverall cardinality across both the first and second Bloom filterarrays, the overall cardinality indicative of a total number of uniqueindividuals corresponding to the first and second users who accessed themedia.
 16. The method of claim 15, wherein the first array of firstelements has a same length as the second array of second elements, thelength corresponding to an odd number of elements.
 17. The method ofclaim 15, further including: generating a third array of third elementsbased on a bit-wise union of the first array and the second array, thebit-wise union based on modulo 2 addition; and estimating the overallcardinality based on the third array.
 18. A method comprising: accessinga plurality of Bloom filter arrays including a first Bloom filter array,the first Bloom filter array generated by a first computer of a firstdatabase proprietor, the first Bloom filter array representative offirst users who accessed media, the first users registered with thefirst database proprietor, the first Bloom filter array including afirst array of first elements, values of respective ones of the firstelements being either a 0 or a 1 based on whether quantities of thefirst users allocated to the respective ones of the first elements areeven or odd, ones of the plurality of Bloom filter arrays representativeof different users who accessed the media; estimating, by executing aninstruction with a processor, a first cardinality for the first Bloomfilter array, the first cardinality indicative of a total number of thefirst users who accessed the media; generating a plurality of arraysbased on bit-wise unions between different sets of at least two of theplurality of Bloom filter arrays, the bit-wise unions based on modulo 2addition, ones of the plurality of arrays representative of exclusive-orgroupings of the different users included within datasets underlyingrespective ones of the plurality of Bloom filter arrays; estimating aplurality of exclusive-or cardinalities for the plurality of arrays; andestimating an overall cardinality across the plurality of Bloom filterarrays based on a summation of the exclusive-or cardinalities.
 19. Theapparatus of claim 1, wherein the communications interface is to receivea plurality of Bloom filter arrays including at least one of the firstBloom filter array or the second Bloom filter array, ones of the Bloomfilter arrays representative of different users who accessed the media,the Bloom filter array analyzer to: generate a plurality of arrays basedon bit-wise unions between different sets of at least two of theplurality of Bloom filter arrays, the bit-wise unions based on modulo 2addition, ones of the plurality of arrays representative of exclusive-orgroupings of the different users included within datasets underlyingrespective ones of the plurality of Bloom filter arrays; estimate aplurality of exclusive-or cardinalities for the plurality of arrays; andestimate an overall cardinality across the plurality of Bloom filterarrays based on a summation of the exclusive-or cardinalities.
 20. Theapparatus of claim 1, wherein the first array of first elements has asame length as the second array of second elements.
 21. The apparatus ofclaim 3, wherein the Bloom filter array analyzer is to: determine acount of the first elements with a value of 1; and estimate the overallcardinality based on the count.
 22. The apparatus of claim 3, whereinthe Bloom filter array analyzer is to: determine a multiplicativeconstant based on a noise parameter, the noise parameter defining aprobability at which ones of the values of respective ones of the firstelements are flipped between 0 and 1 independent of an allocation of thefirst users to the respective ones of the first elements; and estimatethe overall cardinality based on the multiplicative constant.
 23. Theapparatus of claim 3, wherein the communications interface is to receivea plurality of Bloom filter arrays including the first Bloom filterarray and the second Bloom filter array, ones of the Bloom filter arraysrepresentative of different users who accessed the media, the Bloomfilter array analyzer to: generate a plurality of arrays based onbit-wise unions between different sets of at least two of the pluralityof Bloom filter arrays, the bit-wise unions based on modulo 2 addition,ones of the plurality of arrays representative of exclusive-or groupingsof the different users included within datasets underlying respectiveones of the plurality of Bloom filter arrays; estimate a plurality ofexclusive-or cardinalities for the plurality of arrays; and estimate theoverall cardinality across the plurality of Bloom filter arrays based ona summation of the exclusive-or cardinalities.
 24. The apparatus ofclaim 6, wherein different ones of the plurality of Bloom filter arraysare associated with a same array length.
 25. The non-transitory computerreadable medium of claim 7, wherein the instructions are to cause themachine to: access a plurality of Bloom filter arrays including at leastone of the first Bloom filter array or the second Bloom filter array,ones of the plurality of Bloom filter arrays representative of differentusers who accessed the media; generate a plurality of arrays based onbit-wise unions between different sets of at least two of the pluralityof Bloom filter arrays, the bit-wise unions based on modulo 2 addition,ones of the plurality of arrays representative of exclusive-or groupingsof the different users included within datasets underlying respectiveones of the plurality of Bloom filter arrays; estimate a plurality ofexclusive-or cardinalities for the plurality of arrays; and estimate anoverall cardinality across the plurality of Bloom filter arrays based ona summation of the exclusive-or cardinalities.
 26. The non-transitorycomputer readable medium of claim 7, wherein the first array of firstelements has a same length as the second array of second elements. 27.The non-transitory computer readable medium of claim 9, wherein theinstructions are to cause the machine to: determine a count of the firstelements with a value of 1; and estimate the overall cardinality basedon the count.
 28. The non-transitory computer readable medium of claim9, wherein the instructions are to cause the machine to: determine amultiplicative constant based on a noise parameter, the noise parameterdefining a probability at which ones of the values of respective ones ofthe first elements are flipped between 0 and 1 independent of anallocation of the first users to the respective ones of the firstelements; and estimate the overall cardinality based on themultiplicative constant.
 29. The non-transitory computer readable mediumof claim 9, wherein the instructions are to cause the machine to: accessa plurality of Bloom filter arrays including the first Bloom filterarray and the second Bloom filter array, ones of the plurality of Bloomfilter arrays representative of different users who accessed the media;generate a plurality of arrays based on bit-wise unions betweendifferent sets of at least two of the plurality of Bloom filter arrays,the bit-wise unions based on modulo 2 addition, ones of the plurality ofarrays representative of exclusive-or groupings of the different usersincluded within datasets underlying respective ones of the plurality ofBloom filter arrays; estimate a plurality of exclusive-or cardinalitiesfor the plurality of arrays; and estimate the overall cardinality acrossthe plurality of Bloom filter arrays based on a summation of theexclusive-or cardinalities.
 30. The non-transitory computer readablemedium of claim 12, wherein different ones of the plurality of Bloomfilter arrays are associated with a same array length.
 31. The method ofclaim 13, further including: accessing a plurality of Bloom filterarrays including at least one of the first Bloom filter array or thesecond Bloom filter array, ones of the Bloom filter arraysrepresentative of different users who accessed the media; generating aplurality of arrays based on bit-wise unions between different sets ofat least two of the plurality of Bloom filter arrays, the bit-wiseunions based on modulo 2 addition, ones of the plurality of arraysrepresentative of exclusive-or groupings of the different users includedwithin datasets underlying respective ones of the plurality of Bloomfilter arrays; estimating a plurality of exclusive-or cardinalities forthe plurality of arrays; and estimating an overall cardinality acrossthe plurality of Bloom filter arrays based on a summation of theexclusive-or cardinalities.
 32. The method of claim 13, wherein thefirst array of first elements has a same length as the second array ofsecond elements.
 33. The method of claim 15, further including:determining a count of the first elements with a value of 1; andestimating the overall cardinality based on the count.
 34. The method ofclaim 15, further including: determining a multiplicative constant basedon a noise parameter, the noise parameter defining a probability atwhich ones of the values of respective ones of the first elements areflipped between 0 and 1 independent of an allocation of the first usersto the respective ones of the first elements; and estimating the overallcardinality based on the multiplicative constant.
 35. The method ofclaim 15, further including: accessing a plurality of Bloom filterarrays including the first Bloom filter array and the second Bloomfilter array, ones of the plurality of Bloom filter arraysrepresentative of different users who accessed the media; generating aplurality of arrays based on bit-wise unions between different sets ofat least two of the plurality of Bloom filter arrays, the bit-wiseunions based on modulo 2 addition, ones of the plurality of arraysrepresentative of exclusive-or groupings of the different users includedwithin datasets underlying respective ones of the plurality of Bloomfilter arrays; estimating a plurality of exclusive-or cardinalities forthe plurality of arrays; and estimating the overall cardinality acrossthe plurality of Bloom filter arrays based on a summation of theexclusive-or cardinalities.
 36. The method of claim 18, whereindifferent ones of the plurality of Bloom filter arrays are associatedwith a same array length.